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Chapter 15 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

Answers to End-of-Chapter Questions for Chapter 15

Solutions Manual to Accompany The Statistical Imagination:

Elementary Statistics for the Social Sciences by Ferris J. Ritchey

2 edition, McGraw-Hill, 2008/2009nd

by

Brian P. Hinote, Jason Wasserman and Ferris J. Ritchey

Questions for Chapter 15 are on pages 575-576 of the text.

1. For Pearson’s r correlation coefficient, draw a conceptual diagram depicting a population and a sample and insert the appropriate statistics and parameters. Stipulate how the null hypothesis is stated for the hypothesis test.

Research question: Are persons high in religiosity less tolerant of ethnic diversity? That is, the more religious a person, the less likely he or she has an accepting attitude toward new ethnic groups coming into their community.

The null hypothesis is stated as no relationship between the two variables. We know that if no relationship exists, then D (Pearson’s r coefficient for the entire population) would calculate to zero.

0 H : D = 0. (adults)

That is, there is no relationship between religiosity and ethnic tolerance.

Chapter 15 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

2. For a hypothesis test of the relationship between two interval/ratio variables, what is the shape of the sampling distribution and what test statistic is used?

The sampling distribution is the approximately normal t-distribution and the test statistic is:

3. In testing a hypothesis between two interval/ratio variables, the direction of the test is stated in the alternative hypothesis. For the general case of Y regressed on X, show how the alternative hypothesis is stated when the direction is hypothesized to be positive. Indicate the shape of the pattern of coordinates of a scatterplot when this is the case. Do the same for a negative relationship.

For a positive relationship, the statement of the alternative hypothesis is that rho is greater than zero:

A H : D > 0 (a population)

That is, there is a positive relationship between X and Y. One-tailed.

On a scatterplot, the X,Y-coordinates will fit a cigar-shaped pattern sloping upward. The value of Pearson’s r will be positive and significantly greater than zero.

For a negative relationship, the statement of the alternative hypothesis is that rho is less than zero:

A H : D < 0 (a population)

That is, there is a negative relationship between X and Y. One-tailed.

On a scatterplot, the X,Y-coordinates will fit a cigar-shaped pattern sloping downward. The value of Pearson’s r will be negative and significantly less than zero.

Chapter 15 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

4. Observe the formulas for the Pearson’s r correlation coefficient and the regression coefficient b and explain why these two coefficients always have the same directional sign (+ or -).

The formulas are as follows:

For both formulas the numerators are the same. Moreover, in both formulas the denominators will always calculate to a positive value because negative numbers are squared away. This means the sign of the quotient is determined by the numerator, and the signs are the same for both formulas.

5. Explain what r ² tells us.

r = the proportion of the variation in Y explained by knowing that it is related to X.2

The variation is the sum of the squared deviations of Y-scores from the mean of Y. Mathematically, with correlation and regression we are attempting to explain why some subjects in a population score above the mean of Y, while others score below. If X is related to Y, then X explains some part of these deviation scores. For example, why is John so heavy (i.e., above the mean weight)? Is it partly because he is also above average on height (X )? With an especially strong relationship, X provides information to very precisely predict Y using the regression formula.

6. In ascertaining the strength of a relationship between two interval/ratio variables, what is the danger in relying solely on an interpretation of the absolute value of Pearson’s r ? Explain.

Observing r directly (i.e., without squaring) encourages an overestimation of the strength of the relationship. For instance, directly observing, say, r = .50 could lead to the incorrect conclusion that we are “half way there” in reducing errors in prediction. In fact, this correlation is only a quarter of the way there because r = (.50) = .25; that is, only 252 2

percent of the variation in Y is explained by X.

7. Provide a one-word answer to this question: When no relationship is found between X and Y, what do we say about the other aspects of a relationship?

Nothing.

Chapter 15 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

8. Match the following with regard to the relationship between two interval/ratio variables:

a. Pearson’s r The Y-intercept; the value of Y when X = 0

b. a The proportion of variation in Y explained by knowledge of X; a measure of the strength of the relationship.

c. r ² Slope of the regression line; the effect on Y of a one-unit change in X; a measure of the practical applications of the relationship

d. b Predicted value of Y; best estimate of Y for a given value of X; used to describe the practical applications of the relationship

e. Measures how tightly X,Y-coordinates fit around the regression line; used to describe the existence and direction of a relationship

Answers from top to bottom: b, c, d, e, a

9. Explain the difference between cross-sectional data and longitudinal data.

Cross-sectional data are collected at one point in time for each person in a sample. Such a sample is called a cross-sectional sample. For example, a survey conducted on a random sample of 1,000 students between March 1 and March 10 is a cross-sectional sample. Although the survey took 10 days, each respondent was interviewed only once. Data collected over a period of time with multiple contacts with subjects are called longitudinal data and the sample is called a longitudinal sample. For example, in a study of adjustment to college life, we might interview a random sample of first-year students every month for two terms. The interpretation of data is different for a longitudinal sample than for a cross- sectional sample.

10. Using cross-sectional data, a researcher finds a negative relationship between age and knowledge about personal computers. Does his mean that as a people age, they lose this knowledge? Explain.

No. This cross-sectional sample, one taken at a single point in time. We have no way to determine if someone is losing knowledge from one point in time to another because we examined each subject just once. The correlation reveals that, as you compare younger subjects to older subjects, there is a tendency for knowledge about personal computers to diminish progressively though the sample. In other words, older generations tend to know less about personal computers. The correlation does not apply to what happens to a single individual over time.

Chapter 15 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

11. Mathematically, there is a positive correlation between shoe size and the ability to do complicated mathematical problems. Is this really meaningful? Explain.

No. This is a spurious correlation. Mathematically, there is a positive correlation. The larger the shoe size, the greater the ability to solve math problems. What we have actually established, however, is that adults are better at solving math problems than are children. We have indirectly measured age by way of shoe size; the older an individual, in general, the larger the foot and shoe. Foot and shoe size per se do not influence mathematical abilities. The spurious nature of this correlation would be apparent if we correlated shoe size and ability to solve mathematical problems while holding age constant, say, at 30 years. This correlation would not be statistically different from zero. A mathematical correlation by itself proves nothing. To be meaningful, it must make practical sense and/or have the support of a theoretical argument.

12. Mathematically, there is a negative correlation between the number of movies being produced in Hollywood each year and the size of the Amazon rain forests. Is this really meaningful? Explain.

No. This is a spurious correlation. Mathematically, there is a negative correlation. As time has passed, there has been a steady increase in the number of movies being produced in Hollywood. Similarly, as time has passed, the size of the Amazon rain forests has diminished steadily. But this mathematical correlation by itself proves nothing. Many things are gradually increasing or decreasing. Correlating their measurements would produce correlations. To be meaningful, however, a mathematical correlation must make practical sense and/or have the support of a theoretical argument. This example also highlights the importance of predicting correlations and their directions before observing data. Chances are, no one would have hypothesized a relationship between movie making and depletion of rain forests.

END OF ANSWERS TO CHAPTER 15 QUESTIONS