stats homework problems

1. A person's metabolic rate is the rate at which the body consumes energy. Metabolic rate is important in studies of weight gain, dieting, and exercise. Here are the metabolic rates

of 7 men who took part in a study of dieting. (The units are calories per 24 hours. These

are the same calories used to describe the energy content of foods.)

1754 1630 1346 1614 1460 1867 1438

Calculate the mean and standard deviation of the metabolic rates, showing each step in detail.

First find the mean

x

by summing the 7 observations and dividing by 7.

Then find each of the deviations

xi − x

and their squares. What is the sum of the deviations?

Calculate the variance as an average of the squared deviations (remember to divide by

n − 1).

Finally, obtain s as the square root of the variance. (Round your answer to two decimal places.)

2. Click on the median for each of the following graphs. (One of these lines is the mean.)

3. The Wechsler Adult Intelligence Scale (WAIS) is the most common "IQ test." The scale of

scores is set separately for each age group and is approximately normal with mean 100 and

standard deviation 15. (Enter your answer to two decimal places.)

(a) People with WAIS scores below 70 are considered mentally retarded when, for example,

applying for Social Security disability benefits. What percent of adults are retarded by this

criterion? (Round your answer to two decimal places.)

%

(b) The organization MENSA, which calls itself "the high IQ society," requires WAIS scores of

130 or higher for membership. What percent of adults would qualify for membership? (Round

your answer to two decimal places.)

%

4. The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75.

The interquartile range IQR is the distance between the first and third quartiles of a distribution.

(a) What is the value of the IQR for the standard Normal distribution? (Round your answer to

four decimal places.)

(b) There is a constant c such that

IQR = cσ for any Normal distribution

N(μ, σ). What is the value of c? (Round your answer to four decimal places.)

5.

Raw scores on behavioral tests are often transformed for easier comparison. A test of reading

ability has mean 74 and standard deviation 5 when given to third graders. Sixth graders have

mean score 82 and standard deviation 7 on the same test. To provide separate "norms" for each

grade, we want scores in each grade to have mean 100 and standard deviation 20. (Round your

answers to two decimal places.)

(a) What linear transformation will change third-grade scores x into new scores xnew = a + bx that

have the desired mean and standard deviation? (Use b > 0 to preserve the order of the scores.)

a =

b =

(b) Do the same for the sixth-grade scores.

a =

b =

(c) David is a third-grade student who scores 79 on the test. Find David's transformed score.

Nancy is a sixth-grade student who scores 79. What is her transformed score?

Who scores higher within his or her grade?

David Nancy

6. A study of reading ability in schoolchildren chose 60 fifth-grade children at random from a

school.† The main purpose of this study was to ask whether schoolchildren can estimate their

own reading ability. The researchers had the children's scores on a test of reading ability. They

asked each child to estimate his or her reading level, on a scale from 1 (low) to 5 (high). The

figure shown below is a scatterplot of the children's estimates (response) against their reading

scores (explanatory).

(a) What explains the "stair-step" pattern in the plot?

The response variable (estimated level) can only take on the values 1, 2, 3, 4, 5, so the points

in the scatterplot must fall on one of those five levels. Children are very good at estimating

their reading ability. The relationship between the explanatory variable (child's score) and

the response variable (estimated level) will always exhibit this pattern. Many relationships in

the real world exhibit this pattern naturally.

(b) Is there an overall positive association between reading score and self-estimate?

The association is (strongly) negative. The association is (weakly) negative. The

association is (strongly) positive. The association is (weakly) positive.

(c) There is one clear outlier. What is this child's self-estimated reading level?

Does this appear to over- or underestimate the level as measured by the test?

overestimate underestimate

7. Each of the following statements contains a blunder. Explain in each case what is wrong.

(a) "There is a high correlation between the gender of American workers and their income."

Income of American workers can not be correlated with other variables because it is too

complex. Gender would not have any association with American workers income.

The relationship between gender and American workers income has a curved relationship and

correlation measures the strength of only the linear relationship between two variables.

Because gender has a categorical (nominal) scale, we cannot compute the correlation between

sex and anything.

(b) "We found a high correlation

(r = 1.06)

between students' ratings of faculty teaching and ratings made by other faculty members."

A correlation r = 1.06 is impossible because

−1 ≤ r ≤ 1

always. The relationship between students' ratings of faculty teaching and ratings made by

other faculty members would have more of a curved relationship and correlation measures the

strength of only the linear relationship between two variables. Because students' ratings of

faculty teaching has a categorical (nominal) scale, we cannot compute the correlation between

these ratings and anything. The correlation between students' ratings of faculty teaching and

ratings made by other faculty members would not be this high.

(c) "The correlation between planting rate and yield of corn was found to be

r = 0.26

bushel."

Because planting rate has a categorical (nominal) scale, we cannot compute the correlation

between planting rate and anything. There is no correlation between planting rate and yield

of corn. Correlation has no units, so r = 0.26 bushel is incorrect. The correlation

between planting rate and yield of corn would be much higher.

8. The data from data201.dat contains information on 78 seventh-grade students. We want to

know how well each of IQ score and self-concept score predicts GPA using least-squares

regression. We also want to know which of these explanatory variables predicts GPA better.

Give numerical measures that answer these questions. (Round your answers to three decimal

places.)

(Regressor: IQ) R 2

(Regressor: Self-Concept) R 2

Which variable is the better predictor?

IQ Self Concept

9. A study of class attendance and grades among first-year students at a state university showed that in general students who attended a higher percent of their classes earned higher grades. Class attendance explained 36% of the variation in grade index among the students. What is the numerical value of the correlation between percent of classes attended and grade index? (Enter your answer to one decimal place.)

10. Some statistical methods require that the residuals from a regression line have a normal

distribution. Consider the following residuals.

0.37 -0.70 0.10 -0.34 0.19 0.61 -0.26 -0.98

1.64 -0.18 -0.23 0.54 -0.54 -1.11 0.93 -0.03

Is their distribution close to normal? Make a normal quantile plot to find out. (Do this on paper.

Your instructor may ask you to turn this in.)

The distribution appears skewed to the left. The distribution appears skewed to the

right. The distribution appears approximately normal. We can not make any inference

about the shape of the distribution.

11. Every few years, the National Assessment of Educational Progress asks a national sample of

eighth-graders to perform the same math tasks. The goal is to get an honest picture of progress in

math. Suppose these are the last few national mean scores, on a scale of 0 to 500.

Year 1990 1992 1996 2000 2003 2005 2008

Score 264 266 270 271 276 277 279

(a) Make a time plot of the mean scores, by hand. This is just a scatterplot of score against year.

There is a slow linear increasing trend. (Do this on your own.)

(b) Find the regression line of mean score on time step-by-step. First calculate the mean and

standard deviation of each variable and their correlation (use a calculator with these functions).

Then find the equation of the least-squares line from these. (Round your answers to two decimal

places.)

= + x

Draw the line on your scatterplot. What percent of the year-to-year variation in scores is

explained by the linear trend? (Round your answer to one decimal place.)

%

(c) Now use software or the regression function on your calculator to verify your regression line.

(Do this on your own.)

12. A student wonders if tall women tend to date taller men than do short women. She measures

herself, her dormitory roommate, and the women in the adjoining rooms; then she measures the

next man each woman dates. The data (heights in inches) are listed below.

Women (x) 63 63 66 63 69 63

Men (y) 70 68 68 70 70 68

(a) Make a scatterplot of these data. (Do this on paper. Your instructor may ask you to turn this

in.) Based on the scatterplot, do you expect the correlation to be positive or negative? Near ± 1 or not?

The correlation should be positive. It should be near 1. The correlation should be

negative. It should be near -1. The correlation should be positive. It should not be near 1.

The correlation should be negative. It should not be near -1.

(b) Find the correlation r between the heights of the men and women. (Round your answer to

three decimal places.)

(c) How would r change if all the men were 6 inches shorter than the heights given in the table?

Does the correlation tell us whether women tend to date men taller than themselves?

r will not change. This correlation does imply that women tend to date men taller than

themselves. r will increase. This correlation does not imply that women tend to date men

taller than themselves. r will not change. This correlation does not imply that women tend

to date men taller than themselves. r will decrease. This correlation does imply that women

tend to date men taller than themselves.

(d) If heights were measured in centimeters rather than inches, how would the correlation

change? (There are 2.54 centimeters in an inch.)

r will decrease by 2.54. r will increase by 2.54. r will not change. r will be

multiplied by 2.54. r will be divided by 2.54.

(e) If every woman dated a man exactly 3 inches taller than herself, what would be the

correlation between male and female heights?

r will be equal to -1. r will be equal to 1. r will be the same as the value in (b).

r will be equal to 0.

13. The table below gives data from a study that shows that social exclusion causes "real pain."

That is, activity in the area of the brain that responds to physical pain goes up as distress from

social exclusion goes up. The scatterplot for this data shows a moderately strong linear

relationship. The data are given below.

Social Brain

Social Brain

Subject distress activity Subject distress activity

1

1.22

−0.055 8

2.22

0.024

2

1.81

−0.040 9

2.60

0.026

3

1.04

−0.025 10

2.75

0.033

4

2.48

−0.018 11

2.75

0.064

5

2.19

−0.018 12

3.31

0.077

6

2.65

0.018 13

3.63

0.124

7

2.01

0.021

(a) What is the equation of the least-squares regression line for predicting brain activity from

social distress score? (Round your answers to four decimal places.)

brain activity = social distress +

Make a scatterplot with this line drawn on it.

(b) On your plot, show the "up and over" lines that predict brain activity for social distress score

2.4. Use the equation of the regression line to get the predicted brain activity level. Verify that it

agrees with your plot. (Round your answer to four decimal places.)

(c) What percent of the variation in brain activity among these subjects is explained by the

straight-line relationship with social distress score? (Round your answer to a whole number.)

%

14. Find the mean and standard deviation of the times and icicle lengths for the data on run 8903

in data data85.dat. Find the correlation between the two variables. Use these five numbers to find

the equation of the regression line for predicting length from time. Use the same five numbers to

find the equation of the regression line for predicting the time an icicle has been growing from its

length. (Round your answers to three decimal places.)

times x =

times s =

lengths x =

lengths s =

r =

time = + length

length = + time