Chapter 2

1. Jason appears to be growing slowly as a toddler. His height between 18 and 30

months of age increases as follows:

Age (mos) Height (cm)

18 76.5

21 78.7

24 82.0

27 84.8

30 86.0

a.) Make a graph of Jason's height against his age. Do the data show a clear linear

pattern so that you are willing to use a fitted line as an overall description?

b.) The least-squares regression line fitted to these data is

y = 61.5 + 0.837t

Plot this line on your graph from (a).

c.) According to this line, how much does Jason grow each month? If this line described

Jason's growth from birth, what would be his height at birth? (In fact, growth is not

linear in the early months of life, so the line does not describe his birth height.)

d.) Use the fitted line to predict Jason's height at age 2 years. Then calculate the residual

for this age. Where do the residuals appear in your graph from (b)?

e.) Would you be willing to use the fitted line to predict Jason's height at 21 years?

Explain your answer.

2. A rural landowner has a pond with area 50,000 square feet. One day he notices a

growth of algae in one corner of the pond covering 8 square feet. The algae doubles

its area each day.

a.) How much area does the algae occupy 10 days later?

b.) At this point the landowner becomes concerned because about 16% of the pond is

covered with algae. How many more days will the algae take to cover the entire

pond?

3. Identify each bold face variable below as quantitative or categorical, and also identify

the explanatory and response variable in each setting.

a.) A political scientist believes that there is a gender gap in American voting, with

women more likely to vote Democratic. She therefore interviews a random sample

of voters and records the sex of the respondents and the political party of the

candidate for whom they voted in the last presidential election.

b.) A study of the relationship between education and income records the annual

earned income and the years of school completed of each of a large sample of

subjects. In addition, the study records the type of high school attended (public,

religious, or private non-religious).

4. The following table gives data on the lean body mass (kilograms) and resting

metabolic rate for 12 women and 7 men who are subjects in a study of obesity. The

researchers suspect that lean body mass (that is, the subject's weight leaving out all

fat) is an important influence on metabolic rate.

Subject Sex Mass Rate

_________________________________________

1 M 62.0 1792

2 M 62.9 1666

3 F 36.1 995

4 F 54.6 1425

5 F 48.5 1396

6 F 42.0 1418

7 M 47.4 1362

8 F 50.6 1502

9 F 42.0 1256

10 M 48.7 1614

11 F 40.3 1189

12 F 33.1 913

13 M 51.9 1460

14 F 42.4 1124

15 F 34.5 1052

16 F 51.1 1347

17 F 41.2 1204

18 M 51.9 1867

19 M 46.9 1439

a.) Make a scatterplot of the data for the female subjects. Which is the explanatory variable?

b.) Is the association between these variables positive or negative? What is the overall shape of the

relationship?

c.) Now add the data for the male subjects to your graph, using a different color or a different plotting

symbol. Does the type of relationship that you observed in (b) hold for men also? How do the male

subjects as a group differ from the female subjects as a group?

5. Many manatees in Florida are killed or injured by power boats. The table below gives

data on power boat registration (in thousands) and the number of manatees killed by

boats in Florida in the years 1977-1987.

a.) Make a scatterplot of boat registrations and manatees killed. The overall relationship is roughly linear.

b.) Calculate the least-squares regression line and draw it on your graph.

c.) Power boat registrations in 1990 increased to 719,000. Based on the data given here, predict the

number of manatees killed by boats in 1990.

d.) Which point on the graph has the largest residual, either positive or negative? Calculate the residual for

that point. Do you think that this point will be highly influential?

Manatees

Year Boats Killed

___________________________________

1977 447 13

1978 460 21

1979 481 24

1980 498 16

1981 513 24

1982 512 20

1983 526 15

1984 559 34

1985 585 33

1986 614 33

1987 645 39

When adults are asked their weight, the weight that they report tends to be less than

their actual weight as measured by a scale. But there is a strong relationship between

reported weight and measured weight, because heavy people usually report higher

weights than do light people. Here are the measured weights x and reported weights

y (in pounds) for 5 female subjects.

______________________

x 112 123 178 141 135

_______________________

y 110 120 165 125 129

________________________

6. Make a scatterplot of these data. Which observation has the greatest influence on the

position of the regression line and the value of the correlation coefficient?

7. Compute the correlation coefficient r between x and y. What percent of the variation

in the weights reported by these subjects is accounted for by the fact that reported

weight varies linearly with measured weight?

8. Suppose that all of the subjects reported a weight 5 pounds less than the values of y in

the table. Would this change the value of r?

9. Use your value of the correlation r and the means and standard deviations of x and y to

give the equation of the least-squares regression line of reported weight on measured

weight. Draw this line on your scatterplot.

10. Explain what is wrong with each of the following statements.

a.) "Our study shows that the correlation between a voter's religion and the political

party he or she prefers is r = 0.45."

b.) "We found that the correlation between the number of hours per week a student

spends watching television and the student's grades is r = - 1.13."