Southeast Washington state has a long history of seasonal dust storms. Several researchers decided to see what effect, if any, these storms had on the respiratory health of the people living in the area. They undertook (among other things) to see if there was a relationship between the amount of dust and sand particles in the air when the storms occur and the number of hospital emergency room visits for respiratory disorders at three community hospitals in southeast Washington. Using methods of correlation and regression, which are explained in this chapter, they were able to determine the effect of these dust storms on local residents. See Statistics Today—Revisited at the end of the chapter.

Source: B. Hefflin, B. Jalaludin, N. Cobb, C. Johnson, L. Jecha, and R. Etzel, “Surveillance for Dust Storms and Respiratory Diseases in Washington State, 1991,” Archives of Environmental Health 49, no. 3 (May-June 1994), pp. 170–74. Reprinted with permission of the Helen Dwight Reid Education Foundation. Published by Heldref Publications, 1319 18th St. N.W., Washington, D.C. 20036–1802. Copyright 1994.

Introduction

In Chapters 7 and 8 , two areas of inferential statistics—confidence intervals and hypothesis testing—were explained. Another area of inferential statistics involves determining whether a relationship exists between two or more numerical or quantitative variables. For example, a businessperson may want to know whether the volume of sales for a given month is related to the amount of advertising the firm does that month. Educators are interested in determining whether the number of hours a student studies is related to the student’s score on a particular exam. Medical researchers are interested in questions such as, Is caffeine related to heart damage? or Is there a relationship between a person’s age and his or her blood pressure? A zoologist may want to know whether the birth weight of a certain animal is related to its life span. These are only a few of the many questions that can be answered by using the techniques of correlation and regression analysis. Correlation is a statistical method used to determine whether a relationship between variables exists. Regression is a statistical method used to describe the nature of the relationship between variables, that is, positive or negative, linear or nonlinear.

The purpose of this chapter is to answer these questions statistically:

1.Are two or more variables related?

2.If so, what is the strength of the relationship?

3.What type of relationship exists?

4.What kind of predictions can be made from the relationship?

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Unusual Stat

A person walks on average 100,000 miles in his or her lifetime. This is about 3.4 miles per day.

To answer the first two questions, statisticians use a numerical measure to determine whether two or more variables are related and to determine the strength of the relationship between or among the variables. This measure is called a correlation coefficient. For example, there are many variables that contribute to heart disease, among them lack of exercise, smoking, heredity, age, stress, and diet. Of these variables, some are more important than others; therefore, a physician who wants to help a patient must know which factors are most important.

To answer the third question, you must ascertain what type of relationship exists. There are two types of relationships: simple and multiple. In a simple relationship, there are two variables—an independent variable, also called an explanatory variable or a predictor variable, and a dependent variable, also called a response variable. A simple relationship analysis is called simple regression, and there is one independent variable that is used to predict the dependent variable. For example, a manager may wish to see whether the number of years the salespeople have been working for the company has anything to do with the amount of sales they make. This type of study involves a simple relationship, since there are only two variables—years of experience and amount of sales.

In a multiple relationship, called multiple regression, two or more independent variables are used to predict one dependent variable. For example, an educator may wish to investigate the relationship between a student’s success in college and factors such as the number of hours devoted to studying, the student’s GPA, and the student’s high school background. This type of study involves several variables.

Simple relationships can also be positive or negative. A positive relationship exists when both variables increase or decrease at the same time. For instance, a person’s height and weight are related; and the relationship is positive, since the taller a person is, generally, the more the person weighs. In a negative relationship, as one variable increases, the other variable decreases, and vice versa. For example, if you measure the strength of people over 60 years of age, you will find that as age increases, strength generally decreases. The word generally is used here because there are exceptions.

Finally, the fourth question asks what type of predictions can be made. Predictions are made in all areas and daily. Examples include weather forecasting, stock market analyses, sales predictions, crop predictions, gasoline price predictions, and sports predictions. Some predictions are more accurate than others, due to the strength of the relationship. That is, the stronger the relationship is between variables, the more accurate the prediction is.

Paired data and scatter plots were explained in Section 2–4 ; however, a brief review is included here.

Objective 1

Draw a scatter plot for a set of ordered pairs.

10–1Correlation

In simple correlation and regression studies, the researcher collects data on two numerical or quantitative variables to see whether a relationship exists between the variables. For example, if a researcher wishes to see whether there is a relationship between number of hours of study and test scores on an exam, she must select a random sample of students, determine the hours each studied, and obtain their grades on the exam. A table can be made for the data, as shown here.

Student	Hours of study x	Grade y (%)
A	6	82
B	2	63
C	1	57
D	5	88
E	2	68
F	3	75

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As stated previously, the two variables for this study are called the independent variable and the dependent variable. The independent variable is the variable in regression that can be controlled or manipulated. In this case, the number of hours of study is the independent variable and is designated as the x variable. The dependent variable is the variable in regression that cannot be controlled or manipulated. The grade the student received on the exam is the dependent variable, designated as the y variable. The reason for this distinction between the variables is that you assume that the grade the student earns depends on the number of hours the student studied. Also, you assume that, to some extent, the student can regulate or control the number of hours he or she studies for the exam.

The determination of the x and y variables is not always clear-cut and is sometimes an arbitrary decision. For example, if a researcher studies the effects of age on a person’s blood pressure, the researcher can generally assume that age affects blood pressure. Hence, the variable age can be called the independent variable, and the variable blood pressure can be called the dependent variable. On the other hand, if a researcher is studying the attitudes of husbands on a certain issue and the attitudes of their wives on the same issue, it is difficult to say which variable is the independent variable and which is the dependent variable. In this study, the researcher can arbitrarily designate the variables as independent and dependent.

The independent and dependent variables can be plotted on a graph called a scatter plot. The independent variable x is plotted on the horizontal axis, and the dependent variable y is plotted on the vertical axis.

A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable x and the dependent variable y.

The scatter plot is a visual way to describe the nature of the relationship between the independent and dependent variables. The scales of the variables can be different, and the coordinates of the axes are determined by the smallest and largest data values of the variables.

The procedure for drawing a scatter plot is shown in Examples 10–1 .

Example 10–1

Car Rental Companies

Construct a scatter plot for the data shown for car rental companies in the United States for a recent year.

Company	Cars (in ten thousands)	Revenue (in billions)
A	63.0	$7.0
B	29.0	3.9
C	20.8	2.1
D	19.1	2.8
E	13.4	1.4
F	8.5	1.5

Source: Auto Rental News.

Solution

Step 1Draw and label the x and y axes.

Step 2Plot each point on the graph, as shown in Figure 10–1 .

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Figure 10–1

Scatter Plot for Example 10–1

Objective 2

Compute the correlation coefficient.

Correlation

As stated in the Introduction, statisticians use a measure called the correlation coefficient to determine the strength of the linear relationship between two variables. There are several types of correlation coefficients. The one explained in this section is called the Pearson product moment correlation coefficient (PPMC), named after statistician Karl Pearson, who pioneered the research in this area.

The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables. The symbol for the sample correlation coefficient is r. The symbol for the population correlation coefficient is ρ (Greek letter rho).

The range of the correlation coefficient is from –1 to +1. If there is a strong positive linear relationship between the variables, the value of r will be close to +1. If there is a strong negative linear relationship between the variables, the value of r will be close to –1. When there is no linear relationship between the variables or only a weak relationship, the value of r will be close to 0. See Figure 10–2 .

The graphs in Figure 10–3 show the relationship between the correlation coefficients and their corresponding scatter plots. Notice that as the value of the correlation coefficient increases from 0 to +1 (parts a, b, and c), data values become closer to an increasingly stronger relationship. As the value of the correlation coefficient decreases from 0 to –1 (parts d, e, and f), the data values also become closer to a straight line. Again this suggests a stronger relationship.

There are several ways to compute the value of the correlation coefficient. One method is to use the formula shown on the next page.

Figure 10–2

Range of Values for the Correlation Coefficient

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Figure 10–3

Relationship Between the Correlation Coefficient and the Scatter Plot

Formula for the Correlation Coefficient r

where n is the number of data pairs.

Rounding Rule for the Correlation CoefficientRound the value of r to three decimal places.

The formula looks somewhat complicated, but using a table to compute the values, as shown in Example 10–2 , makes it somewhat easier to determine the value of r.

There are no units associated with r, and the value of r will remain unchanged if the x and y values are switched.

Example 10–2

Car Rental Companies

Compute the correlation coefficient for the data in Example 10–1 .

Solution

Step 1Make a table as shown here.

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Step 2Find the values of xy, x2, and y2 and place these values in the corresponding columns of the table.

The completed table is shown.

Step 3Substitute in the formula and solve for r.

The correlation coefficient suggests a strong relationship between the number of cars a rental agency has and its annual income.

Correlation and CausationResearchers must understand the nature of the linear relationship between the independent variable x and the dependent variable y. When a hypothesis test indicates that a significant linear relationship exists between the variables, researchers must consider the possibilities outlined next.

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Possible Relationships Between Variables

When the null hypothesis has been rejected for a specific a value, any of the following five possibilities can exist.

1.There is a direct cause-and-effect relationship between the variables. That is, x causes y. For example, water causes plants to grow, poison causes death, and heat causes ice to melt.

2.There is a reverse cause-and-effect relationship between the variables. That is, y causes x. For example, suppose a researcher believes excessive coffee consumption causes nervousness, but the researcher fails to consider that the reverse situation may occur. That is, it may be that an extremely nervous person craves coffee to calm his or her nerves.

3.The relationship between the variables may be caused by a third variable. For example, if a statistician correlated the number of deaths due to drowning and the number of cans of soft drink consumed daily during the summer, he or she would probably find a significant relationship. However, the soft drink is not necessarily responsible for the deaths, since both variables may be related to heat and humidity.

4.There may be a complexity of interrelationships among many variables. For example, a researcher may find a significant relationship between students’ high school grades and college grades. But there probably are many other variables involved, such as IQ, hours of study, influence of parents, motivation, age, and instructors.

5.The relationship may be coincidental. For example, a researcher may be able to find a significant relationship between the increase in the number of people who are exercising and the increase in the number of people who are committing crimes. But common sense dictates that any relationship between these two values must be due to coincidence.

When two variables are highly correlated, item 3 in the box states that there exists a possibility that the correlation is due to a third variable. If this is the case and the third variable is unknown to the researcher or not accounted for in the study, it is called a lurking variable. An attempt should be made by the researcher to identify such variables and to use methods to control their influence.

It is important to restate the fact that even if the correlation between two variables is high, it does not necessarily mean causation. There are other possibilities, such as lurking variables or just a coincidental relationship. See the Speaking of Statistics article on page 540 .

Also, you should be cautious when the data for one or both of the variables involve averages rather than individual data. It is not wrong to use averages, but the results cannot be generalized to individuals since averaging tends to smooth out the variability among individual data values. The result could be a higher correlation than actually exists.

Thus, when the null hypothesis is rejected, the researcher must consider all possibilities and select the appropriate one as determined by the study. Remember, correlation does not necessarily imply causation.

Objective 3

Test the hypothesis H0: ρ = 0.

The Significance of the Correlation CoefficientThis section requires coverage of Chapter 8 . As stated before, the range of the correlation coefficient is between –1 and +1. When the value of r is near +1 or –1, there is a strong linear relationship. When the value of r is near 0, the linear relationship is weak or nonexistent. Since the value of r is computed from data obtained from samples, there are two possibilities when r is not equal to zero: either the value of r is high enough to conclude that there is a significant linear relationship between the variables, or the value of r is due to chance.

To make this decision, you use a hypothesis-testing procedure. The traditional method is similar to the one used in previous chapters.

Step 1State the hypotheses.

Step 2Find the critical values.

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Step 3Compute the test value.

Step 4Make the decision.

Step 5Summarize the results.

The population correlation coefficient is computed from taking all possible (x, y) pairs; it is designated by the Greek letter ρ (rho). The sample correlation coefficient can then be used as an estimator of ρ if the following assumptions are valid.

1.The variables x and y are linearly related.

2.The variables are random variables.

3.The two variables have a bivariate normal distribution.

A biviarate normal distribution means that for the pairs of (x, y) data values, the corresponding y values have a bell-shaped distribution for any given x value, and the x values for any given y value have a bell-shaped distribution.

Formally defined, the population correlation coefficient ρ is the correlation computed by using all possible pairs of data values (x, y) taken from a population.

Interesting Fact

Scientists think that a person is never more than 3 feet away from a spider at any given time!

In hypothesis testing, one of these is true:

H0: ρ = 0	This null hypothesis means that there is no correlation between the x and y variables in the population.
H1: ρ ≠ 0	This alternative hypothesis means that there is a significant correlation between the variables in the population.

When the null hypothesis is rejected at a specific level, it means that there is a significant difference between the value of r and 0. When the null hypothesis is not rejected, it means that the value of r is not significantly different from 0 (zero) and is probably due to chance.

Several methods can be used to test the significance of the correlation coefficient. Three methods will be shown in this section. The first uses the t test.

Historical Note

A mathematician named Karl Pearson (1857–1936) became interested in Francis Galton’s work and saw that the correlation and regression theory could be applied to other areas besides heredity. Pearson developed the correlation coefficient that bears his name.

Formula for the t Test for the Correlation Coefficient

with degrees of freedom equal to n – 2.

Although hypothesis tests can be one-tailed, most hypotheses involving the correlation coefficient are two-tailed. Recall that ρ represents the population correlation coefficient. Also, if there is no linear relationship, the value of the correlation coefficient will be 0. Hence, the hypotheses will be

H0: ρ = 0andH1: ρ ≠ 0

You do not have to identify the claim here, since the question will always be whether there is a significant linear relationship between the variables.

The two-tailed critical values are used. These values are found in Table F in Appendix C . Also, when you are testing the significance of a correlation coefficient, both variables x and y must come from normally distributed populations.

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Example 10–3

Test the significance of the correlation coefficient found in Example 10–2 . Use α = 0.05 and r = 0.982.

Solution

Step 1State the hypotheses.

H0: ρ = 0andH1. ρ ≠ 0

Step 2Find the critical values. Since α = 0.05 and there are 6 – 2 = 4 degrees of freedom, the critical values obtained from Table F are ±2.776, as shown in Figure 10–4 .

Figure 10–4

Critical Values for Example 10–3

Step 3Compute the test value.

Step 4Make the decision. Reject the null hypothesis, since the test value falls in the critical region, as shown in Figure 10–5 .

Figure 10–5

Test Value for Example 10–3

Step 5Summarize the results. There is a significant relationship between the number of cars a rental agency owns and its annual income.

The second method that can be used to test the significance of r is the R-value method. The method is the same as that shown in Chapters 8 and 9 . It uses the following steps.

Step 1State the hypotheses.

Step 2Find the test value. (In this case, use the t test.)

Step 3Find the P-value. (In this case, use Table F .)

Step 4Make the decision.

Step 5Summarize the results.

Consider an example where t = 4.059 and d.f. = 4. Using Table F with d.f. = 4 and the row Two tails, the value 4.059 falls between 3.747 and 4.604; hence, 0.01 < P-value < 0.02. (The P-value obtained from a calculator is 0.015.) That is, the P-value falls between 0.01 and 0.02. The decision, then, is to reject the null hypothesis since P-value < 0.05.

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The third method of testing the significance of r is to use Table I in Appendix C . This table shows the values of the correlation coefficient that are significant for a specific α level and a specific number of degrees of freedom. For example, for 7 degrees of freedom and α = 0.05, the table gives a critical value of 0.666. Any value of r greater than +0.666 or less than –0.666 will be significant, and the null hypothesis will be rejected. See Figure 10–6 . When Table I is used, you need not compute the t test value. Table I is for two-tailed tests only.

Figure 10–6

Finding the Critical Value from Table I

Example 10–4

Using Table I , test the significance of the correlation coefficient r = 0.067 at α = 0.01 when the sample size is 9.

Solution

H0: ρ = 0andH1: ρ ≠ 0

Since the sample size is 9, there are 7 degrees of freedom. When α = 0.01 and with 7 degrees of freedom, the value obtained from Table I is 0.798. For a significant relationship, a value of r greater than +0.798 or less than –0.798 is needed. Since r = 0.067, the null hypothesis is not rejected. Hence, there is not enough evidence to say that there is a significant linear relationship between the variables. See Figure 10–7 .

Figure 10–7

Rejection and Nonrejection Regions for Example 10–4

Applying the Concepts 10–1

Stopping Distances

In a study on speed control, it was found that the main reasons for regulations were to make traffic flow more efficient and to minimize the risk of danger. An area that was focused on in the study was the distance required to completely stop a vehicle at various speeds. Use the following table to answer the questions.

MPH	Braking distance (feet)
20	20
30	45
40	81
50	133
60	205
80	411

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Assume MPH is going to be used to predict stopping distance.

1.Which of the two variables is the independent variable?

2.Which is the dependent variable?

3.What type of variable is the independent variable?

4.What type of variable is the dependent variable?

5.Construct a scatter plot for the data.

6.Is there a linear relationship between the two variables?

7.Redraw the scatter plot, and change the distances between the independent-variable numbers. Does the relationship look different?

8.Is the relationship positive or negative?

9.Can braking distance be accurately predicted from MPH?

10.List some other variables that affect braking distance.

11.Compute the value of r.

12.Is r significant at α = 0.05?

See page 569 for the answers.

Speaking of Statistics

In correlation and regression studies, it is difficult to control all variables. This study shows some of the consequences when researchers overlook certain aspects in studies. Suggest ways that the extraneous variables might be controlled in future studies.

Coffee Not Disease Culprit, Study Says

NEW YORK (AP)—Two new studies suggest that coffee drinking, even up to 5½ cups per day, does not increase the risk of heart disease, and other studies that claim to have found increased risks might have missed the true culprits, a researcher says.

“It might not be the coffee cup in one hand, it might be the cigarette or coffee roll in the other,” said Dr. Peter W. F. Wilson, the author of one of the new studies.

He noted in a telephone interview Thursday that many coffee drinkers, particularly heavy coffee drinkers, are smokers. And one of the new studies found that coffee drinkers had excess fat in their diets.

The findings of the new studies conflict sharply with a study reported in November 1985 by Johns Hopkins University scientists in Baltimore.

The Hopkins scientists found that coffee drinkers who consumed five or more cups of coffee per day had three times the heart- disease risk of non-coffee drinkers.

The reason for the discrepancy appears to be that many of the coffee drinkers in the Hopkins study also smoked—and it was the smoking that increased their heart-disease risk, said Wilson.

Wilson, director of laboratories for the Framingham Heart Study in Framingham, Mass., said Thursday at a conference sponsored by the American Heart Association in Charleston, S.C., that he had examined the coffee intake of 3,937 participants in the Framingham study during 1956–66 and an additional 2,277 during the years 1972–1982.

In contrast to the subjects in the Hopkins study, most of these coffee drinkers consumed two or three cups per day, Wilson said. Only 10 percent drank six or more cups per day.

He then looked at blood cholesterol levels and heart and blood vessel disease in the two groups. “We ran these analyses for coronary heart disease, heart attack, sudden death and stroke and in absolutely every analysis, we found no link with coffee,” Wilson said.

He found that coffee consumption was linked to a significant decrease in total blood cholesterol in men, and to a moderate increase in total cholesterol in women.

Source: Reprinted with permission of the Associated Press.

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Exercises 10–1

1.What is meant by the statement that two variables are related?

2.How is a linear relationship between two variables measured in statistics? Explain.

3.What is the symbol for the sample correlation coefficient? The population correlation coefficient?

4.What is the range of values for the correlation coefficient?

5.What is meant when the relationship between the two variables is called positive? Negative?

6.Give examples of two variables that are positively correlated and two that are negatively correlated.

7.Give an example of a correlation study, and identify the independent and dependent variables.

8.What is the diagram of the independent and dependent variables called? Why is drawing this diagram important?

9.What is the name of the correlation coefficient used in this section?

10.What statistical test is used to test the significance of the correlation coefficient?

11.When two variables are correlated, can the researcher be sure that one variable causes the other? Why or why not?

For Exercises 12 through 27, perform the following steps.

a.Draw the scatter plot for the variables.

b.Compute the value of the correlation coefficient.

c.State the hypotheses.

d.Test the significance of the correlation coefficient at α = 0.05, using Table I .

e.Give a brief explanation of the type of relationship.

12. Broadway Productions A researcher wants to see if there is a relationship between the number of new productions on Broadway in any given year and the attendance for the season. The data below were recorded for a selected number of years. Based on these data, can you conclude a relationship between the number of new productions in a season and the attendance?

(The information in this exercise will be used for Exercise 12 in Section 10–2 .)

Source: World Almanac.

13.Commercial Movie Releases The yearly data have been published showing the number of releases for each of the commercial movie studios and the gross receipts for those studios thus far. Based on these data, can it be concluded that there is a relationship between the number of releases and the gross receipts?

(The information in this exercise will be used for Exercises 13 and 36 in Section 10–2 and Exercises 15 and 19 in Section 10–3 .)

Source: www.showbizdata.com

14. Forest Fires and Acres Burned An environmentalist wants to determine the relationships between the numbers (in thousands) of forest fires over the year and the number (in hundred thousands) of acres burned. The data for 8 recent years are shown. Describe the relationship.

Source: National Interagency Fire Center.

(The information in this exercise will be used for Exercise 14 in Section 10–2 and Exercises 16 and 20 in Section 10–3 .)

15. Alumni Contributions The director of an alumni association for a small college wants to determine whether there is any type of relationship between the amount of an alumnus’s contribution (in dollars) and the years the alumnus has been out of school. The data follow. (The information is used for Exercises 15, 36, and 37 in Section 10–2 and Exercises 17 and 21 in Section 10–3 .)

16.State Debt and Per Capita Tax An economics student wishes to see if there is a relationship between the amount of state debt per capita and the amount of tax per capita at the state level. Based on the following data, can she or he conclude that per capita state debt and per capita state taxes are related? Both amounts are in dollars and represent five randomly selected states. (The information in this exercise will be used for Exercises 16 and 37 in Section 10–2 and Exercises 18 and 22 in Section 10–3 .)

Source: World Almanac.

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17. Larceny and Vandalism A criminology student wishes to see if there is a relationship between the number of larceny crimes and the number of vandalism crimes on college campuses in southwestern Pennsylvania. The data are shown. Is there a relationship between the two types of crimes?

(The information in this exercise will be used for Exercise 17 of Section 10–2 .)

18. Pass Attempts A football fan wishes to see how the number of pass attempts (not completions) relates to the number of yards gained for quarterbacks in past NFL season playoff games. The data are shown for five quarterbacks. Describe the relationships.

(The information in this exercise will be used for Exercises 18 and 38 in Section 10–2 .)

19. Egg Production Recent agricultural data showed the number of eggs produced and the price received per dozen for a given year. Based on the following data for a random selection of states, can it be concluded that a relationship exists between the number of eggs produced and the price per dozen? (The information in this exercise will be used for Exercise 19 in Section 10–2 .)

Source: World Almanac.

20. Emergency Calls and Temperature An emergency service wishes to see whether a relationship exists between the outside temperature and the number of emergency calls it receives for a 7-hour period. The data are shown. (The information in this exercise will be used for Exercises 20 and 38 in Section 10–2 .)

21. Distribution of Population in U.S. Cities A random sample of U.S. cities is selected to determine if there is a relationship between the population (in thousands) of people under 5 years of age and the population (in thousands) of those 65 years of age and older. The data for the sample are shown here. (The information in this exercise will be used for Exercises 21 and 36 in Section 10–2 .)(The information in this exercise will be used for Exercise 17 of Section 10–2 .)

Source: New York Times Almanac.

22.Apartment Rents The results of a survey of the average monthly rents (in dollars) for existing one- bedroom and three-bedroom apartments in randomly selected metropolitan areas are shown below. Determine if there is a significant relationship between the rents. (The information in this exercise will be used for Exercise 22 in Section 10–2 .)

Source: New York Times Almanac.

23. Average Temperature and Precipitation The average normal daily temperature (in degrees Fahrenheit) and the corresponding average monthly precipitation (in inches) for the month of June are shown here for seven randomly selected cities in the United States. Determine if there is a relationship between the two variables. (The information in this exercise will be used for Exercise 23 in Section 10–2 .)

Source: New York Times Almanac.

24. Hall of Fame Pitchers A random sample of Hall of Fame pitchers’ career wins and their total number of strikeouts is shown next. Is there a relationship between the variables? (The information in this exercise will be used for Exercise 24 in Section 10–2 .)

Source: New York Times Almanac.

25. Calories and Cholesterol The number of calories and the number of milligrams of cholesterol for a random sample of fast-food chicken sandwiches from seven restaurants are shown here. Is there a relationship between the variables? (The information in this exercise will be used in Exercise 25 in Section 10–2 .)

Source: The Doctor’s Pocket Calorie, Fat, and Carbohydrate Counter.

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26.Tall Buildings An architect wants to determine the relationship between the heights (in feet) of a building and the number of stories in the building. The data for a sample of 10 buildings in Pittsburgh are shown. Explain the relationship.

Source: World Almanac Book of Facts.

(The information in this exercise will be used for Exercise 26 of Section 10–2 .)

27.Hospital Beds A hospital administrator wants to see if there is a relationship between the number of licensed beds and the number of staffed beds in local hospitals. The data for a specific day are shown. Describe the relationship.

Source: Pittsburgh Tribune-Review.

(The information in this exercise will be used for Exercise 28 of this section and Exercise 27 in Section 10–2 .)

Extending the Concepts

28.One of the formulas for computing r is

Using the data in Exercise 27, compute r with this formula. Compare the results.

29. Compute r for the data set shown. Explain the reason ’ for this value of r. Now, interchange the values of x and y and compute r again. Compare this value with the previous one. Explain the results of the comparison.

30. Compute r for the following data and test the hypothesis H0: ρ = 0. Draw the scatter plot; then explain the results.

Objective 4

Compute the equation of the regression line.

10–2Regression

In studying relationships between two variables, collect the data and then construct a scatter plot. The purpose of the scatter plot, as indicated previously, is to determine the nature of the relationship. The possibilities include a positive linear relationship, a negative linear relationship, a curvilinear relationship, or no discernible relationship. After the scatter plot is drawn, the next steps are to compute the value of the correlation coefficient and to test the significance of the relationship. If the value of the correlation coefficient is significant, the next step is to determine the equation of the regression line, which is the data’s line of best fit. (Note: Determining the regression line when r is not significant and then making predictions using the regression line are meaningless.) The purpose of the regression line is to enable the researcher to see the trend and make predictions on the basis of the data.

Line of Best Fit

Figure 10–8 shows a scatter plot for the data of two variables. It shows that several lines can be drawn on the graph near the points. Given a scatter plot, you must be able to draw the line of best fit. Best fit means that the sum of the squares of the vertical distances from each point to the line is at a minimum. The reason you need a line of best fit is that the values of y will be predicted from the values of x; hence, the closer the points are to the line, the better the fit and the prediction will be. See Figure 10–9 . When r is positive, the line slopes upward and to the right. When r is negative, the line slopes downward from left to right.

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Figure 10–8

Scatter Plot with Three Lines Fit to the Data

Figure 10–9

Line of Best Fit for a Set of Data Points

Historical Note

Francis Galton drew the line of best fit visually. An assistant of Karl Pearson’s named G. Yule devised the mathematical solution using the least-squares method, employing a mathematical technique developed by Adrien-Marie Legendre about 100 years earlier.

Determination of the Regression Line Equation

In algebra, the equation of a line is usually given as y = mx + b, where m is the slope of the line and b is the y intercept. (Students who need an algebraic review of the properties of a line should refer to Appendix A , Section A–3, before studying this section.) In statistics, the equation of the regression line is written as y' = a + bx, where a is the y' intercept and b is the slope of the line. See Figure 10–10 .

There are several methods for finding the equation of the regression line. Two formulas are given here. These formulas use the same values that are used in computing the value of the correlation coefficient. The mathematical development of these formulas is beyond the scope of this book.

Formulas for the Regression Line y' = a + bx

where a is the y' intercept and b is the slope of the line.

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Figure 10–10

A Line as Represented in Algebra and in Statistics

Rounding Rule for the Intercept and SlopeRound the values of a and b to three decimal places.

Example 10–5

Car Rental Companies

Find the equation of the regression line for the data in Example 10–4 , and graph the line on the scatter plot of the data.

Solution

The values needed for the equation are n = 6, Σx = 153.8, Σy = 18.7, Σxy = 682.77, and Σx2 = 5859.26. Substituting in the formulas, you get

Hence, the equation of the regression line y' = a + bx is

y' = 0.396 + 0.106x

To graph the line, select any two points for x and find the corresponding values for y. Use any x values between 10 and 60. For example, let x = 15. Substitute in the equation and find the corresponding y' value.

y' = 0.396

= 0.396 + 0.106(15)

= 1.986

Let x = 40; then

y' = 0.396 + 0.106x

= 0.396 + 0.106(40)

= 4.636

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Figure 10–11

Regression Line for Example 10–5

Historical Note

In 1795, Adrien-Marie Legendre (1752–1833) measured the meridian arc on the earth’s surface from Barcelona, Spain, to Dunkirk, England. This measure was used as the basis for the measure of the meter. Legendre developed the least-squares method around the year 1805.

Then plot the two points (15, 1.986) and (40, 4.636) and draw a line connecting the two points. See Figure 10–11 .

Note: When you draw the regression line, it is sometimes necessary to truncate the graph (see Chapter 2 ). This is done when the distance between the origin and the first labeled coordinate on the x axis is not the same as the distance between the rest of the labeled x coordinates or the distance between the origin and the first labeled y' coordinate is not the same as the distance between the other labeled y' coordinates. When the x axis or the y axis has been truncated, do not use the y' intercept value to graph the line. When you graph the regression line, always select x values between the smallest x data value and the largest x data value.

The sign of the correlation coefficient and the sign of the slope of the regression line will always be the same. That is, if r is positive, then b will be positive; if r is negative, then b will be negative. The reason is that the numerators of the formulas are the same and determine the signs of r and b, and the denominators are always positive. The regression line will always pass through the point whose x coordinate is the mean of the x values and whose y coordinate is the mean of the y values, that is, .

The regression line can be used to make predictions for the dependent variable. The method for making predictions is shown in Example 10–6 .

Example 10–6

Car Rental Companies

Use the equation of the regression line to predict the income of a car rental agency that has 200,000 automobiles.

Solution

Since the x values are in 10,000s, divide 200,000 by 10,000 to get 20, and then substitute 20 for x in the equation.

y' = 0.396 + 0.106x

= 0.396 + 0.106(20)

= 2.516

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Hence, when a rental agency has 200,000 automobiles, its revenue will be approximately $2.516 billion.

The value obtained in Example 10–6 is a point prediction, and with point predictions, no degree of accuracy or confidence can be determined. More information on prediction is given in Section 10–3 .

The magnitude of the change in one variable when the other variable changes exactly 1 unit is called a marginal change. The value of slope b of the regression line equation represents the marginal change. For example, in Example 10–5 the slope of the regression line is 0.106, which means for each increase of 10,000 cars, the value of y changes 0.106 unit ($106 million) on average.

When r is not significantly different from 0, the best predictor of y is the mean of the data values of y. For valid predictions, the value of the correlation coefficient must be significant. Also, two other assumptions must be met.

Assumptions for Valid Predictions in Regression

1.For any specific value of the independent variable x, the value of the dependent variable y must be normally distributed about the regression line. See Figure 10–12(a) .

2.The standard deviation of each of the dependent variables must be the same for each value of the independent variable. See Figure 10–12(b) .

Figure 10–12

Assumptions for Predictions

Extrapolation, or making predictions beyond the bounds of the data, must be interpreted cautiously. For example, in 1979, some experts predicted that the United States would run out of oil by the year 2003. This prediction was based on the current consumption and on known oil reserves at that time. However, since then, the automobile industry has produced many new fuel-efficient vehicles. Also, there are many as yet undiscovered oil fields. Finally, science may someday discover a way to run a car on something as unlikely but as common as peanut oil. In addition, the price of a gallon of gasoline was predicted to reach $10 a few years later. Fortunately this has not come to pass. Remember that when predictions are made, they are based on present conditions or on the premise that present trends will continue. This assumption may or may not prove true in the future.

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The steps for finding the value of the correlation coefficient and the regression line equation are summarized in this Procedure Table:

Interesting Fact

It is estimated that wearing a motorcycle helmet reduces the risk of a fatal accident by 30%.

Procedure Table

Finding the Correlation Coefficient and the Regression Line Equation

Step 1Make a table, as shown in step 2.

Step 2Find the values of xy, x2, and y2. Place them in the appropriate columns and sum each column.

Step 3Substitute in the formula to find the value of r.

Step 4When r is significant, substitute in the formulas to find the values of a and b for the regression line equation y' = α + bx.

A scatter plot should be checked for outliers. An outlier is a point that seems out of place when compared with the other points (see Chapter 3 ). Some of these points can affect the equation of the regression line. When this happens, the points are called influential points or influential observations.

When a point on the scatter plot appears to be an outlier, it should be checked to see if it is an influential point. An influential point tends to “pull” the regression line toward the point itself. To check for an influential point, the regression line should be graphed with the point included in the data set. Then a second regression line should be graphed that excludes the point from the data set. If the position of the second line is changed considerably, the point is said to be an influential point. Points that are outliers in the x direction tend to be influential points.

Researchers should use their judgment as to whether to include influential observations in the final analysis of the data. If the researcher feels that the observation is not necessary, then it should be excluded so that it does not influence the results of the study. However, if the researcher feels that it is necessary, then he or she may want to obtain additional data values whose x values are near the x value of the influential point and then include them in the study.

Source: Reprinted with special permission of King Features Syndicate.

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Applying the Concepts 10–2

Stopping Distances Revisited

In a study on speed and braking distance, researchers looked for a method to estimate how fast a person was traveling before an accident by measuring the length of the skid marks. An area that was focused on in the study was the distance required to completely stop a vehicle at various speeds. Use the following table to answer the questions.

MPH	Braking distance (feet)
20	20
30	45
40	81
50	133
60	205
80	411

Assume MPH is going to be used to predict stopping distance.

1.Find the linear regression equation.

2.What does the slope tell you about MPH and the braking distance? How about the y intercept?

3.Find the braking distance when MPH = 45.

4.Find the braking distance when MPH = 100.

5.Comment on predicting beyond the given data values.

See page 569 for the answers.

Exercises 10–2

1.What two things should be done before one performs a regression analysis?

2.What are the assumptions for regression analysis?

3.What is the general form for the regression line used in statistics?

4.What is the symbol for the slope? For the y intercept?

5.What is meant by the line of best fit?

6.When all the points fall on the regression line, what is the value of the correlation coefficient?

7.What is the relationship between the sign of the correlation coefficient and the sign of the slope of the regression line?

8.As the value of the correlation coefficient increases from 0 to 1, or decreases from 0 to –1, how do the points of the scatter plot fit the regression line?

9.How is the value of the correlation coefficient related to the accuracy of the predicted value for a specific value of x?

10.If the value of r is not significant, what can be said about the regression line?

11.When the value of r is not significant, what value should be used to predict y?

For Exercises 12 through 27, use the same data as for the corresponding exercises in Section 10–1 . For each exercise, find the equation of the regression line and find the y' value for the specified x value. Remember that no regression should be done when r is not significant.

12.Broadway Productions New Broadway productions and seasonal attendance are as follows.

Find y' when x = 48 new productions.

13.Commercial Movie Releases New movie releases per studio and gross receipts are as follows:

Find y' when x = 200 new releases.

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14.Forest Fires and Acres Burned Number of fires and number of acres burned are as follows:

Find y' when x = 60.

15.Years and contribution data are as follows:

Find y' when x = 4 years.

16.State Debt and Per Capita Taxes Data for per capita state debt and per capita state tax are as follows:

Find y' when x = $1500 in per capita debt.

17.Larceny and Vandalism Larceny crimes and vandalism crimes are as follows:

Find y' when x = 40.

18.Pass Attempts Data for pass attempts and yards gained are as follows:

Find y' when x = 95.

19.Egg Production Number of eggs and price per dozen are shown.

Find y' when x = 1600 million eggs.

20.Emergency Calls and Temperature Temperature in degrees Fahrenheit and number of emergency calls are shown.

Find y' when x = 80°F.

21.Distribution of Population in U.S. Cities Number (in thousands) of people under 5 years old and people 65 and over living in six randomly selected cities in the United States are shown.

Find y' when x = 200 thousand.

22.Apartment Rents Data for one-bedroom apartment rent and three-bedroom apartment rent are as follows:

Find y' when x = $700 per month.

23.Average Temperature and Precipitation Temperatures (in degrees Fahrenheit) and precipitation (in inches) are as follows:

Find y' when x = 70°F.

24.Hall of Fame Pitchers Wins and strikeouts for Hall of Fame pitchers data are as follows:

Find y' when x = 260 wins.

25.Calories and Cholesterol Calories and cholesterol are as follows:

Find y' when x = 600 calories.

26.Tall Buildings Stories and heights of buildings data follow:

Find y' when x = 44.

27.Hospital Beds Licensed beds and staffed beds data follow:

Find y' when x = 44.

For Exercises 28 through 33, do a complete regression analysis by performing these steps.

a.Draw a scatter plot.

b.Compute the correlation coefficient.

c.State the hypotheses.

d.Test the hypotheses at α = 0.05. Use Table I .

e.Determine the regression line equation.

f.Plot the regression line on the scatter plot.

g.Summarize the results.

28. Fireworks and Injuries These data were obtained for the years 1993 through 1998 and indicate the number of fireworks (in millions) used and the related injuries. Predict the number of injuries if 100 million fireworks are used during a given year.

Source: National Council of Fireworks Safety, American Pyrotechnic Assoc.

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29.Farm Acreage Is there a relationship between the number of farms in a state and the acreage per farm? A random selection of states across the country, both eastern and western, produced the following results. Can a relationship between these two variables be concluded?

Source: World Almanac.

30.SAT Scores Educational researchers desired to find out if a relationship exists between the average SAT verbal score and the average SAT mathematical score. Several states were randomly selected, and their SAT average scores are recorded below. Is there sufficient evidence to conclude a relationship between the two scores?

Source: World Almanac.

31. Coal Production These data were obtained from a sample of counties in southwestern Pennsylvania and indicate the number (in thousands) of tons of bituminous coal produced in each county and the number of employees working in coal production in each county. Predict the number of employees needed to produce 500 thousand tons of coal. The data are given here.

32.Television Viewers A television executive selects 10 television shows and compares the average number of viewers the show had last year with the average number of viewers this year. The data (in millions) are shown. Describe the relationship.

Source: Nielson Media Research.

33.Absences and Final Grades An educator wants to see how the number of absences for a student in her class affects the student’s final grade. The data obtained from a sample are shown.

For Exercises 34 and 35, do a complete regression analysis and test the significance of r at α = 0.05, using the P-value method.

34. Father’s and Son’s Weights A physician wishes to know whether there is a relationship between a father’s weight (in pounds) and his newborn son’s weight (in pounds). The data are given here.

35.Age and Net Worth Is a person’s age related to his or her net worth? A sample of 10 billionaires is selected, and the person’s age and net worth are compared. The data are given here.

Source: The Associated Press.

Extending the Concepts

36.For Exercises 13, 15, and 21 in Section 10–1 , find the mean of the x and y variables. Then substitute the mean of the x variable into the corresponding regression line equations found in Exercises 13, 15, and 21 in this section and find y'. Compare the value of y' with y for each exercise. Generalize the results.

37.The y intercept value α can also be found by using the equation

Verify this result by using the data in Exercises 15 and 16 of Sections 10–1 and 10–2 .

38.The value of the correlation coefficient can also be found by using the formula

where sx is the standard deviation of the x values and sy is the standard deviation of the y values. Verify this result for Exercises 18 and 20 of Section 10–1 .

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Technology Step by Step

MINITAB

Step by Step

Create a Scatter Plot

1.These instructions use the following data:

Enter the data into three columns. The subject column is optional (see step 6b).

2.Name the columns C1 Subject, C2 Age, and C3 Pressure.

3.Select Graph>Scatterplot, then select Simple and click [OK].

4.Double-click on C3 Pressure for the [Y] variable and C2 Age for the predictor [X] variable.

5.Click [Data View]. The Data Display should be Symbols. If not, click the option box to select it. Click [OK].

6.Click [Labels].

a)Type Pressure vs. Age in the text box for Titles/Footnotes, then type Your Name in the box for Subtitle 1.

b)Optional: Click the tab for Data Labels, then click the option to Use labels from column.

c)Select C1 Subject.

7.Click [OK] twice.

Calculate the Correlation Coefficient

8.Select Stat>Basic Statistics>Correlation.

9.Double-click C3 Pressure, then double-click C2 Age. The box for Display p-values should be checked.

10.Click [OK]. The correlation coefficient will be displayed in the session window, r = +0.897 with a P-value of 0.015.

Determine the Equation of the Least-Squares Regression Line

11.Select Stat>Regression>Regression.

12.Double-click Pressure in the variable list to select it for the Response variable Y.

13.Double-click C2 Age in the variable list to select it for the Predictors variable X.

14.Click on [Storage], then check the boxes for Residuals and Fits.

15.Click [OK] twice.

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The session window will contain the regression analysis as shown.

In the worksheet two new columns will be added with the fitted values and residuals. Summary: The scatter plot and correlation coefficient confirm a strong positive linear correlation between pressure and age. The null hypothesis would be rejected at a significance level of 0.015. The equation of the regression equation is pressure = 81.0 + 0.964 (age).

Regression Analysis: Pressure versus Age

TI–83 Plus or TI–84 Plus

Step by Step

Correlation and Regression

To graph a scatter plot:

1.Enter the x values in L1 and the y values in L2.

2.Make sure the Window values are appropriate. Select an Xmin slightly less than the smallest x data value and an Xmax slightly larger than the largest x data value. Do the same for Ymin and Ymax. Also, you may need to change the Xscl and Yscl values, depending on the data.

3.Press 2nd [STAT PLOT] 1 for Plot 1. The other y functions should be turned off.

4.Move the cursor to On and press ENTER on the Plot 1 menu.

5.Move the cursor to the graphic that looks like a scatter plot next to Type (first graph), and press ENTER. Make sure the X list is L1, and the Y list is L2.

6.Press GRAPH.

Example TI10–1

Draw a scatter plot for the following data.

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The input and output screens are shown.

To find the equation of the regression line:

1.Press STAT and move the cursor to Calc.

2.Press 8 for LinReg(a+bx) then ENTER. The values for a and b will be displayed.

In order to have the calculator compute and display the correlation coefficient and coefficient of determination as well as the equation of the line, you must set the diagnostics display mode to on. Follow these steps:

1.Press 2nd [CATALOG].

2.Use the arrow keys to scroll down to DiagnosticOn.

3.Press ENTER to copy the command to the home screen.

4.Press ENTER to execute the command.

You will have to do this only once. Diagnostic display mode will remain on until you perform a similar set of steps to turn it off.

Example TI10–2

Find the equation of the regression line for the data in Example TI10–1. The input and output screens are shown.

The equation of the regression line is y' = 81.04808549 + 0.964381122x.

To plot the regression line on the scatter plot:

1.Press Y = and CLEAR to clear any previous equations.

2.Press VARS and then 5 for Statistics.

3.Move the cursor to EQ and press 1 for RegEQ. The line will be in the Y= screen.

4.Press GRAPH.

Example TI10–3

Draw the regression line found in Example TI10–2 on the scatter plot.

The output screens are shown.

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To test the significance of b and ρ:

1.Press STAT and move the cursor to TESTS.

2.Press E (ALPHA SIN) for LinRegTTest. Make sure the Xlist is L1, the Ylist is L2, and the Freq is 1. (Use F for TI–84)

3.Select the appropriate alternative hypothesis.

4.Move the cursor to Calculate and press ENTER.

Example TI10–4

Test the hypothesis H0: ρ = 0 for the data in Example TI 10–1. Use α = 0.05.

In this case, the t test value is 4.050983638. The P-value is 0.0154631742, which is significant. The decision is to reject the null hypothesis at α = 0.05, since 0.0154631742 < 0.05; r = 0.8966728145, r2 = 0.8040221364.

There are two other ways to store the equation for the regression line in Y1 for graphing.

1.Type Y1 after the LinReg(a+bx) command.

2.Type Y1 in the RegEQ: spot in the LinRegTTest.

To get Y1 do this:

Press VARS for variables, move cursor to Y-VARS, press 1 for Function, press 1 for Y1.

Excel

Step by Step

Scatter Plots

Creating a scatter plot is straightforward when you use the Chart Wizard.

1.You must have at least two columns of data to use the Scatter Plot option.

2.Highlight the data to be plotted. Select the Insert tab from the toolbar. Then select the Scatter chart and the first type (Scatter with only markers).

3.By left-clicking anywhere on the chart, you automatically bring up the Chart Tools group on the toolbar. The Chart Tools menu includes three additional tabs for editing your chart: Design, Layout, and Format.

4.You can add titles to your chart and to the axes by selecting the Layout tab, then selecting the appropriate option from the Labels group.

Correlation Coefficient

The CORREL function in Excel returns the correlation coefficient without regression analysis.

1.Enter the data in columns A and B.

2.Select a blank cell, and then select the Formulas tab from the toolbar.

3.Select Insert Function icon from the toolbar.

4.Select the Statistical function category and select the CORREL function.

5.Enter the data range A1:AN, where N is the number of sample data pairs for the first variable in Array1. Enter the data range B1:BN for the second variable in Array2, and then click [OK].

Correlation and Regression

This procedure will allow you to calculate the Pearson product moment correlation coefficient without performing a regression analysis.

1.Enter the data from Example 10–2 in a new worksheet. Enter the seven values for the numbers of absences in column A and the corresponding final grades in column B.

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2.Select Data from the toolbar. Then select Data Analysis. Under Analysis Tools, select Correlation.

3.In the Correlation dialog box, type A1:B6 for the Input Range and check the Grouped By: Columns option.

4.Under Output options, select Output Range, and type D2. Then click [OK].

This procedure will allow you to conduct a regression analysis and compute the correlation coefficient. Use the data from Example 10–2 .

1.Select the Data tab on the toolbar, then Data Analysis>Regression.

2.In the Regression dialog box, type B1:B6 in the Input y Range and type A1:A6 in the Input x Range.

3.Under Output options, select Output Range, and type D6. Then click [OK].

Note: To see all of the decimal places for the statistics in the Summary Output, expand the width of columns D to L.

1.Highlight columns D through L.

2.Select the Home tab, and then select Format Autofit Column Width.

10–3Coefficient of Determination and Standard Error of the Estimate

The previous sections stated that if the correlation coefficient is significant, the equation of the regression line can be determined. Also, for various values of the independent variable x, the corresponding values of the dependent variable y can be predicted. Several other measures are associated with the correlation and regression techniques. They include the coefficient of determination, the standard error of the estimate, and the prediction interval. But before these concepts can be explained, the different types of variation associated with the regression model must be defined.

Types of Variation for the Regression Model

Consider the following hypothetical regression model.

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The equation of the regression line is y' = 4.8 + 2.8x, and r = 0.919. The sample y values are 10, 8, 12, 16, and 20. The predicted values, designated by y', for each x can be found by substituting each x value into the regression equation and finding y'. For example, when x = 1,

y' = 4.8 + 2.8x = 4.8 + (2.8)(1) = 7.6

Now, for each x, there is an observed y value and a predicted y' value; for example, when x = 1, y = 10, and y' = 7.6. Recall that the closer the observed values are to the predicted values, the better the fit is and the closer r is to +1 or –1.

The total variation Σ(y – )2 is the sum of the squares of the vertical distances each point is from the mean. The total variation can be divided into two parts: that which is attributed to the relationship of x and y and that which is due to chance. The variation obtained from the relationship (i.e., from the predicted y' values) is Σ(y' – )2 and is called the explained variation. Most of the variations can be explained by the relationship. The closer the value r is to + 1 or –1, the better the points fit the line and the closer Σ(y' – )2 is to Σ(y – )2. In fact, if all points fall on the regression line, Σ(y' – )2 will equal Σ(y – )2, since y' is equal to y in each case.

On the other hand, the variation due to chance, found by Σ(y – y')2, is called the unexplained variation. This variation cannot be attributed to the relationship. When the unexplained variation is small, the value of r is close to +1 or –1. If all points fall on the regression line, the unexplained variation Σ(y – y')2 will be 0. Hence, the total variation is equal to the sum of the explained variation and the unexplained variation. That is,

These values are shown in Figure 10–13 . For a single point, the differences are called deviations. For the hypothetical regression model given earlier, for x = 1 and y = 10, you get y' = 7.6 and y = 13.2.

The procedure for finding the three types of variation is illustrated next.

Step 1Find the predicted y' values.

For x = 1y' = 4.8 + 2.8x = 4.8 + (2.8)(1) = 7.6

For x = 2y'= 4.8 + (2.8)(2) = 10.4

For x = 3y'= 4.8 + (2.8)(3) = 13.2

For x = 4y'= 4.8 + (2.8)(4) = 16.0

For x = 5y'= 4.8 + (2.8)(5) = 18.8

Figure 10–13

Deviations for the Regression Equation

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Unusual Stat

There are 1,929,770, 126,028,800 different color combinations for Rubik’s cube and only one correct solution in which all the colors of the squares on each face are the same.

Hence, the values for this example are as follows:

x	y	y'
1	10	7.6
2	8	10.4
3	12	13.2
4	16	16.0
5	20	18.8

Step 2Find the mean of the y values.

Step 3Find the total variation Σ(y – )2.

(10 – 13.2)2 = 10.24

(8 – 13.2)2 = 27.04

(12 – 13.2)2 = 1.44

(16 – 13.2)2 = 7.84

(20 – 13.2)2 = 46.24

Σ(y – )2 = 92.8

Step 4Find the explained variation Σ(y' – )2.

(7.6 – 13.2)2 = 31.36

(10.4 – 13.2)2 = 7.84

(13.2 – 13.2)2 = 0.00

(16 – 13.2)2 = 7.84

(18.8 – 13.2)2 = 31.36

Σ(y' – )2 = 78.4

Step 5Find the unexplained variation Σ(y – y')2.

(10 – 7.6)2 = 5.76

(8 – 10.4)2 = 5.76

(12 – 13.2)2 = 1.44

(16 – 16)2 = 0.00

(20 – 18.8)2 = 1.44

Σ(y – y')2 = 14.4

Notice that

Total variation =	Explained variation	+	Unexplained variation
92.8 =	78.4	+	14.4

Note: The values (y – y') are called residuals. A residual is the difference between the actual value of y and the predicted value y' for a given x value. The mean of the residuals is always zero. As stated previously, the regression line determined by the formulas in Section 10–2 is the line that best fits the points of the scatter plot. The sum of the squares of the residuals computed by using the regression line is the smallest possible value. For this reason, a regression line is also called a least-squares line.

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Objective 5

Compute the coefficient of determination.

Coefficient of Determination

The coefficient of determination is the ratio of the explained variation to the total variation and is denoted by r2. That is,

For the example, r2 = 78.4/92.8 = 0.845. The term r2 is usually expressed as a percentage. So in this case, 84.5% of the total variation is explained by the regression line using the independent variable.

Another way to arrive at the value for r2 is to square the correlation coefficient. In this case, r = 0.919 and r2 = 0.845, which is the same value found by using the variation ratio.

The coefficient of determination is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable. The symbol for the coefficient of determination is r2.

Of course, it is usually easier to find the coefficient of determination by squaring r and converting it to a percentage. Therefore, if r = 0.90, then r2 = 0.81, which is equivalent to 81%. This result means that 81% of the variation in the dependent variable is accounted for by the variations in the independent variable. The rest of the variation, 0.19, or 19%, is unexplained. This value is called the coefficient of nondetermination and is found by subtracting the coefficient of determination from 1. As the value of r approaches 0, r2 decreases more rapidly. For example, if r = 0.6, then r2 = 0.36, which means that only 36% of the variation in the dependent variable can be attributed to the variation in the independent variable.

Coefficient of Nondetermination

1.00 – r2

Historical Note

Karl Pearson recommended in 1897 that the French government close all its casinos and turn the gambling devices over to the academic community to use in the study of probability.

Objective 6

Compute the standard error of the estimate.

Standard Error of the Estimate

When a y' value is predicted for a specific x value, the prediction is a point prediction. However, a prediction interval about the y' value can be constructed, just as a confidence interval was constructed for an estimate of the population mean. The prediction interval uses a statistic called the standard error of the estimate .

The standard error of the estimate, denoted by sest, is the standard deviation of the observed y values about the predicted y' values. The formula for the standard error of the estimate is

The standard error of the estimate is similar to the standard deviation, but the mean is not used. As can be seen from the formula, the standard error of the estimate is the square root of the unexplained variation—that is, the variation due to the difference of the observed values and the expected values—divided by n – 2. So the closer the observed values are to the predicted values, the smaller the standard error of the estimate will be.

Example 10–7 shows how to compute the standard error of the estimate.

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Example 10–7

Copy Machine Maintenance Costs

A researcher collects the following data and determines that there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y' = 55.57 + 8.13x. Find the standard error of the estimate.

Machine	Age x (years)	Monthly cost y
A	1	$ 62
B	2	78
C	3	70
D	4	90
E	4	93
F	6	103

Solution

Step 1Make a table, as shown.

Step 2Using the regression line equation y' = 55.57 + 8.13x, compute the predicted values y' for each x and place the results in the column labeled y'.

x = 1y' = 55.57 + (8.13)(1) = 63.70

x = 2y' = 55.57 + (8.13)(2) = 71.83

x = 3y' = 55.57 + (8.13)(3) = 79.96

x = 4y' = 55.57 + (8.13)(4) = 88.09

x = 6y' = 55.57 + (8.13)(6) = 104.35

Step 3For each y, subtract y' and place the answer in the column labeled y – y'.

62 – 63.70 =–1.7090 – 88.09 = 1.91

78 – 71.83 = 6.1793 – 88.09 = 4.91

70 – 79.96 =–9.96103 – 104.35 = –1.35

Step 4Square the numbers found in step 3 and place the squares in the column labeled (y – y')2.

Step 5Find the sum of the numbers in the last column. The completed table is shown.

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Step 6Substitute in the formula and find sest.

In this case, the standard deviation of observed values about the predicted values is 6.51.

The standard error of the estimate can also be found by using the formula

Example 10–8

Find the standard error of the estimate for the data for Example 10–7 by using the preceding formula. The equation of the regression line is y' = 55.57 + 8.13x.

Solution

Step 1Make a table.

Step 2Find the product of x and y values, and place the results in the third column.

Step 3Square the y values, and place the results in the fourth column.

Step 4Find the sums of the second, third, and fourth columns. The completed table is shown here.

Step 5From the regression equation y' = 55.57 + 8.13x, a = 55.57, and b = 8.13.

Step 6Substitute in the formula and solve for sest.

This value is close to the value found in Example 10–7 . The difference is due to rounding.

Objective 7

Find a prediction interval.

Prediction Interval

The standard error of the estimate can be used for constructing a prediction interval (similar to a confidence interval) about a y' value.

When a specific value x is substituted into the regression equation, the y' that you get is a point estimate for y. For example, if the regression line equation for the age of a machine and the monthly maintenance cost is y' = 55.57 + 8.13x (Example 10–7), then the predicted maintenance cost for a 3-year-old machine would be y' = 55.57 + 8.13(3), or $79.96. Since this is a point estimate, you have no idea how accurate it is. But you can construct a prediction interval about the estimate. By selecting an α value, you can achieve a (1 – α) · 100% confidence that the interval contains the actual mean of the y values that correspond to the given value of x.

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The reason is that there are possible sources of prediction errors in finding the regression line equation. One source occurs when finding the standard error of the estimate sest. Two others are errors made in estimating the slope and the y' intercept, since the equation of the regression line will change somewhat if different random samples are used when calculating the equation.

Formula for the Prediction Interval about a Value y'

with d.f. = n – 2.

Example 10–9

For the data in Example 10–7 , find the 95% prediction interval for the monthly maintenance cost of a machine that is 3 years old.

Solution

Step 1Find Σx, Σx2, and .

Σx = 20Σx2 = 82

Step 2Find y' for x = 3.

y' = 55.57 + 8.13x

= 55.57 + 8.13(3) = 79.96

Step 3Find sest.

Sest = 6.48

as shown in Example 10–8 .

Step 4Substitute in the formula and solve: tα/2 = 2.776, d.f. = 6 – 2 = 4 for 95%.

Hence, you can be 95% confident that the interval 60.53 < y < 99.39 contains the actual value of y.

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Applying the Concepts 10–3

Interpreting Simple Linear Regression

Answer the questions about the following computer-generated information.

Linear correlation coefficient r = 0.794556

Coefficient of determination = 0.631319

Standard error of estimate = 12.9668

Explained variation = 5182.41

Unexplained variation = 3026.49

Total variation = 8208.90

Equation of regression liney' = 0.725983X + 16.5523

Level of significance = 0.1

Test statistic = 0.794556

Critical value = 0.378419

1.Are both variables moving in the same direction?

2.Which number measures the distances from the prediction line to the actual values?

3.Which number is the slope of the regression line?

4.Which number is the y intercept of the regression line?

5.Which number can be found in a table?

6.Which number is the allowable risk of making a type I error?

7.Which number measures the variation explained by the regression?

8.Which number measures the scatter of points about the regression line?

9.What is the null hypothesis?

10.Which number is compared to the critical value to see if the null hypothesis should be rejected?

11.Should the null hypothesis be rejected?

See pages 569 and 570 for the answers.

Exercises 10–3

1.What is meant by the explained variation? How is it computed?

2.What is meant by the unexplained variation? How is it computed?

3.What is meant by the total variation? How is it computed?

4.Define the coefficient of determination.

5.How is the coefficient of determination found?

6.Define the coefficient of nondetermination.

7.How is the coefficient of nondetermination found?

For Exercises 8 through 13, find the coefficients of determination and nondetermination and explain the meaning of each.

8. r = 0.80

9. r = 0.75

10. r = 0.35

11. r = 0.42

12. r = 0.18

13. r = 0.91

14.Define the standard error of the estimate for regression. When can the standard error of the estimate be used to construct a prediction interval about a value y'?

15.Compute the standard error of the estimate for Exercise 13 in Section 10–1 . The regression line equation was found in Exercise 13 in Section 10–2 .

16.Compute the standard error of the estimate for Exercise 14 in Section 10–1 . The regression line equation was found in Exercise 14 in Section 10–2 .

17.Compute the standard error of the estimate for Exercise 15 in Section 10–1 . The regression line equation was found in Exercise 15 in Section 10–2 .

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18.Compute the standard error of the estimate for Exercise 16 in Section 10–1 . The regression line equation was found in Exercise 16 in Section 10–2 .

19.For the data in Exercises 13 in Sections 10–1 and 10–2 and 15 in Section 10–3 , find the 90% prediction interval when x = 200 new releases.

20.For the data in Exercises 14 in Sections 10–1 and 10–2 and 16 in Section 10–3 , find the 95% prediction interval when x = 60.

21.For the data in Exercises 15 in Sections 10–1 and 10–2 and 17 in Section 10–3 , find the 90% prediction interval when x = 4 years.

22.For the data in Exercises 16 in Sections 10–1 and 10–2 and 18 in Section 10–3 , find the 98% prediction interval when x = 47 years.

Summary

Many relationships among variables exist in the real world. One way to determine whether a relationship exists is to use the statistical techniques known as correlation and regression. The strength and direction of a linear relationship are measured by the value of the correlation coefficient. It can assume values between and including +1 and –1. The closer the value of the correlation coefficient is to +1 or –1, the stronger the linear relationship is between the variables. A value of +1 or –1 indicates a perfect linear relationship. A positive relationship between two variables means that for small values of the independent variable, the values of the dependent variable will be small, and that for large values of the independent variable, the values of the dependent variable will be large. A negative relationship between two variables means that for small values of the independent variable, the values of the dependent variable will be large, and that for large values of the independent variable, the values of the dependent variable will be small.

Relationships can be linear or curvilinear. To determine the shape, you draw a scatter plot of the variables. If the relationship is linear, the data can be approximated by a straight line, called the regression line, or the line of best fit. The closer the value of r is to +1 or –1, the more closely the points will fit the line.

In addition, relationships can be multiple. That is, there can be two or more independent variables and one dependent variable. A coefficient of correlation and a regression equation can be found for multiple relationships, just as they can be found for simple relationships.

The coefficient of determination is a better indicator of the strength of a relationship than the correlation coefficient. It is better because it identifies the percentage of variation of the dependent variable that is directly attributable to the variation of the independent variable. The coefficient of determination is obtained by squaring the correlation coefficient and converting the result to a percentage.

Another statistic used in correlation and regression is the standard error of the estimate, which is an estimate of the standard deviation of the y values about the predicted y' values. The standard error of the estimate can be used to construct a prediction interval about a specific value point estimate y' of the mean of the y values for a given value of x.

Finally, remember that a significant relationship between two variables does not necessarily mean that one variable is a direct cause of the other variable. In some cases this is true, but other possibilities that should be considered include a complex relationship involving other (perhaps unknown) variables, a third variable interacting with both variables, or a relationship due solely to chance.

Source: Cartoon by Bradford Veley, Marquette, Michigan. Reprinted with permission.

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Important Terms

coefficient of determination

correlation

correlation coefficient

dependent variable

extrapolation

independent variable

influential point or observation

least-squares line

lurking variable

marginal change

multiple relationship

negative relationship

Pearson product moment correlation coefficient

population correlation coefficient

positive relationship

standard error of the estimate

Important Formulas

Formula for the correlation coefficient:

Formula for the t test for the correlation coefficient:

The regression line equation:

y' = a + bx where

where

Formula for the standard error of the estimate:

Formula for the prediction interval for a value y':

Review Exercises

For Exercises 1 through 7, do a complete regression analysis by performing the following steps.

a.Draw the scatter plot.

b.Compute the value of the correlation coefficient.

c.Test the significance of the correlation coefficient at α = 0.01, using Table I .

d.Determine the regression line equation.

e.Plot the regression line on the scatter plot.

f.Predict y' for a specific value of x.

1. Passengers and Airline Fares The U.S. Department of Transportation Office of Aviation Analysis provides the weekly average number of passengers per flight and the average one-way fare in dollars for common commercial routes. Randomly selected flights are listed below with the reported data. Is there evidence of a relationship between these two variables?

Flight	Avg. no. of passengers x	Avg. one-way fare y
Pittsburgh–Washington, DC	310	$236
Chicago–Pittsburgh	1388	105
Cincinnati–New York City	750	339
Denver–Phoenix	3019	96
Denver–Los Angeles	2151	176
Houston–Philadelphia	1104	180

Source: www.fedstats.gov

2. Day Care Centers A researcher wishes to determine if there is a relationship between the number of day care centers and the number of group day care homes for counties in Pennsylvania. If there is a significant relationship, predict the number of group care homes a county has if the county has 20 day care centers.

Source: State Department of Public Welfare.

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3. Gasoline and Cigarette Taxes A study was done to compare various common taxes levied by each of the states. Two specific taxes, tax on gasoline and tax on cigarettes, are recorded here for a random selection of states. Can it be concluded that there is a significant relationship between the two? Predict the cigarette tax if the tax on gasoline is 18.4¢.

Source: www.articles.moneycentral.msn.com

4. Driver’s Age and Accidents A study is conducted to determine the relationship between a driver’s age and the number of accidents he or she has over a 1-year period. The data are shown here. (This information will be used for Exercise 8.) If there is a significant relationship, predict the number of accidents of a driver who is 28.

5 . Typing Speed and Word Processing A researcher desires to know whether the typing speed of a secretary (in words per minute) is related to the time (in hours) that it takes the secretary to learn to use a new word processing program. The data are shown.

If there is a significant relationship, predict the time it will take the average secretary who has a typing speed of 72 words per minute to learn the word processing program. (This information will be used for Exercises 9 and 11.)

6. Protein and Diastolic Blood Pressure A study was conducted with vegetarians to see whether the number of grams of protein each ate per day was related to diastolic blood pressure. The data are given here. (This information will be used for Exercises 10 and 12.) If there is a significant relationship, predict the diastolic pressure of a vegetarian who consumes 8 grams of protein per day.

7 . Medical Specialties and Gender Although more and more women are becoming physicians each year, it is well known that men outnumber women in many specialties. Randomly selected specialties are listed below with the numbers of male and female physicians in each. Can it be concluded that there is a significant relationship between the two variables? Predict the number of male specialists when there are 2000 female specialists.

Specialty	Female x	Male y
Dermatology	3,482	6,506
Emergency medicine	5,098	20,429
Neurology	2,895	10,088
Pediatric cardiology	459	1,241
Radiology	1,218	7,574
Forensic pathology	181	399
Radiation oncology	968	3,215

Source: World Almanac.

8.For Exercise 4, find the standard error of the estimate.

9. For Exercise 5, find the standard error of the estimate.

10.For Exercise 6, find the standard error of the estimate.

11. For Exercise 5, find the 90% prediction interval for time when the speed is 72 words per minute.

12.For Exercise 6, find the 95% prediction interval for pressure when the number of grams is 8.

Statistics Today

Do Dust Storms Affect Respiratory Health?–Revisited

The researchers correlated the dust pollutant levels in the atmosphere and the number of daily emergency room visits for several respiratory disorders, such as bronchitis, sinusitis, asthma, and pneumonia. Using the Pearson correlation coefficient, they found overall a significant but low correlation, r = 0.13, for bronchitis visits only. However, they found a much higher correlation value for sinusitis, P-value = 0.08, when pollutant levels exceeded maximums set by the Environmental Protection Agency (EPA). In addition, they found statistically significant correlation coefficients r = 0.94 for sinusitis visits and r = 0.74 for upper-respiratory-tract infection visits 2 days after the dust pollutants exceeded the maximum levels set by the EPA.

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Data Analysis

The Data Bank is found in Appendix D , or on the World Wide Web by following links from www.mhhe.com/bluman/

1.From the Data Bank, choose two variables that might be related: for example, IQ and educational level; age and cholesterol level; exercise and weight; or weight and systolic pressure. Do a complete correlation and regression analysis by performing the following steps. Select a random sample of at least 10 subjects.

a.Draw a scatter plot.

b.Compute the correlation coefficient.

c.Test the hypothesis H0: ρ = 0.

d.Find the regression line equation.

e.Summarize the results.

2.Repeat Exercise 1, using samples of values of 10 or more obtained from Data Set V in Appendix D . Let x = the number of suspensions and y = the enrollment size.

3.Repeat Exercise 1, using samples of 10 or more values obtained from Data Set XIII. Let x = the number of beds and y = the number of personnel employed.

Chapter Quiz

Determine whether each statement is true or false. If the statement is false, explain why.

1. A negative relationship between two variables means that for the most part, as the x variable increases, the y variable increases.

2. A correlation coefficient of –1 implies a perfect linear relationship between the variables.

3. Even if the correlation coefficient is high or low, it may not be significant.

4. When the correlation coefficient is significant, you can assume x causes y.

5. It is not possible to have a significant correlation by chance alone.

6. In multiple regression, there are several dependent variables and one independent variable.

Select the best answer.

7. The strength of the relationship between two variables is determined by the value of

a.r

b.a

c.x

d.sest

8. To test the significance of r, a(n) ______ test is used.

a.t

b.F

c.χ2

d.None of the above

9. The test of significance for r has ______ degrees of freedom.

a.1

b.n

c.n – 1

d.n – 2

10. The equation of the regression line used in statistics is

a.x = a + by

b.y = bx + a

c.y' = a + bx

d.x = ay + b

11. The coefficient of determination is

a.r

b.r2

c.a

d.b

Complete the following statements with the best answer.

12. A statistical graph of two variables is called a(n) _______.

13. The x variable is called the _______ variable.

14. The range of r is from _______ to _______.

15. The sign of r and _______ will always be the same.

16. The regression line is called the _______.

17. If all the points fall on a straight line, the value of r will be _______ or _______.

For Exercises 18 through 21, do a complete regression analysis.

a.Draw the scatter plot.

b.Compute the value of the correlation coefficient.

c.Test the significance of the correlation coefficient at α = 0.05.

d.Determine the regression line equation.

e.Plot the regression line on the scatter plot.

f.Predict y' for a specific value of x.

18 . Prescription Drug Prices A medical researcher wants to determine the relationship between the price per dose of prescription drugs in the United States and the price of the same dose in Australia. The data are shown. Describe the relationship.

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19 . Age and Driving Accidents A study is conducted to determine the relationship between a driver’s age and the number of accidents he or she has over a 1-year period. The data are shown here. If there is a significant relationship, predict the number of accidents of a driver who is 64.

20 . Age and Cavities A researcher desires to know if the age of a child is related to the number of cavities he or she has. The data are shown here. If there is a significant relationship, predict the number of cavities for a child of 11.

21 . Fat and Cholesterol A study is conducted with a group of dieters to see if the number of grams of fat each consumes per day is related to cholesterol level. The data are shown here. If there is a significant relationship, predict the cholesterol level of a dieter who consumes 8.5 grams of fat per day.

22. For Exercise 20, find the standard error of the estimate.

23. For Exercise 21, find the standard error of the estimate.

24. For Exercise 20, find the 90% prediction interval of the number of cavities for a 7-year-old.

25. For Exercise 21, find the 95% prediction interval of the cholesterol level of a person who consumes 10 grams of fat.

Critical Thinking Challenges

Product Sales When the points in a scatter plot show a curvilinear trend rather than a linear trend, statisticians have methods of fitting curves rather than straight lines to the data, thus obtaining a better fit and a better prediction model. One type of curve that can be used is the logarithmic regression curve. The data shown are the number of items of a new product sold over a period of 15 months at a certain store. Notice that sales rise during the beginning months and then level off later on.

1.Draw the scatter plot for the data.

2.Find the equation of the regression line.

3.Describe how the line fits the data.

4.Using the log key on your calculator, transform the x values into log x values.

5.Using the log x values instead of the x values, find the equation of a and b for the regression line.

6.Next, plot the curve y = a + b log x on the graph.

7.Compare the line y = a + bx with the curve y = a + b log x and decide which one fits the data better.

8.Compute r, using the x and y values; then compute r, using the log x and y values. Which is higher?

9.In your opinion, which (the line or the logarithmic curve) would be a better predictor for the data? Why?

Data Projects

Use a significance level of 0.05 for all tests below.

1.Business and Finance Use the stocks in data project 1 of Chapter 2 identified as the Dow Jones Industrials as the sample. For each, note the current price and the amount of the last year’s dividends. Are the two variables linearly related? How much variability in amount of dividend is explainable by the price?

2.Sports and Leisure For each team in major league baseball note the number of wins the team had last year and the number of home runs by its best home run hitter. Is the number of wins linearly related to the number of home runs hit? How much variability in total wins is explained by home runs hit? Write a regression equation to determine how many wins you would expect a team to have, knowing their top home run output.

3.Technology Use the data collected in data project 3 of Chapter 2 for this problem. For the data set note the length of the song and the year it was released. Is there a linear relationship between the length of a song and the year it was released? Is the sign on the correlation coefficient positive or negative? What does the sign on the coefficient indicate about the relationship?

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4.Health and Wellness Use a fast-food restaurant to compile your data. For each menu item note its fat grams and its total calories. Is there a linear relationship between the two variables? How much variance in total calories is explained by fat grams? Write a regression equation to determine how many total calories you would expect in an item, knowing its fat grams.

5.Politics and Economics For each state find its average SAT Math score, SAT English score, and average household income. Which has the strongest linear relationship, SAT Math and SAT English, SAT Math and income, or SAT English and income?

6.Your Class Use the data collected in data project 6 of Chapter 2 regarding heart rates. Is there a linear relationship between the heart rates before and after exercise? How much of the variability in heart rate after exercise is explainable by heart rate before exercise? Write a regression equation to determine what heart rate after exercise you would expect for a person, given the person’s heart rate before exercise.

Answers to Applying the Concepts

Section 10–1 Stopping Distances

1.The independent variable is miles per hour (mph).

2.The dependent variable is braking distance (feet).

3.Miles per hour is a continuous quantitative variable.

4.Braking distance is a continuous quantitative variable.

5.A scatter plot of the data is shown.

6.There might be a linear relationship between the two variables, but there is a bit of a curve in the data.

7.Changing the distances between the mph increments will change the appearance of the relationship.

8.There is a positive relationship between the two variables—higher speeds are associated with longer braking distances.

9.The strong relationship between the two variables suggests that braking distance can be accurately predicted from mph. We might still have some concern about the curve in the data.

10.Answers will vary. Some other variables that might affect braking distance include road conditions, driver response time, and condition of the brakes.

11.The correlation coefficient is r = 0.966.

12.The value for r = 0.966 is significant at α = 0.05. This confirms the strong positive relationship between the variables.

Section 10–2 Stopping Distances Revisited

1.The linear regression equation is y' = –151.90 + 6.4514x

2.The slope says that for each additional mile per hour a car is traveling, we expect the stopping distance to increase by 6.45 feet, on average. The y intercept is the braking distance we would expect for a car traveling 0 mph—this is meaningless in this context, but is an important part of the model.

3. y' = –151.90 + 6.4514(45) = 138.4 The braking distance for a car traveling 45 mph is approximately 138 feet.

4. y' = –151.90 + 6.4514(100) = 493.2 The braking distance for a car traveling 100 mph is approximately 493 feet.

5.It is not appropriate to make predictions of braking distance for speeds outside of the given data values (for example, the 100 mph above) because we know nothing about the relationship between the two variables outside of the range of the data.

Section 10–3 Interpreting Simple Linear Regression

1.Both variables are moving in the same direction. In others words, the two variables are positively associated. This is so because the correlation coefficient is positive.

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2.The unexplained variation of 3026.49 measures the distances from the prediction line to the actual values.

3.The slope of the regression line is 0.725983.

4.The y intercept is 16.5523.

5.The critical value of 0.378419 can be found in a table.

6.The allowable risk of making a type I error is 0.10, the level of significance.

7.The variation explained by the regression is 0.631319, or about 63.1%.

8.The average scatter of points about the regression line is 12.9668, the standard error of the estimate.

9.The null hypothesis is that there is no correlation, H0: ρ = 0.

10.We compare the test statistic of 0.794556 to the critical value to see if the null hypothesis should be rejected.

11.Since 0.794556 > 0.378419, we reject the null hypothesis and find that there is enough evidence to conclude that the correlation is not equal to zero.