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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 12: Q. R12.2 (page 822)

Do taller students require fewer steps to walk a fixed distance? The scatterplot shows the relationship between x=height (in inches) and y=number of steps required to walk the length of a school hallway for a random sample of 36 students at a high school.
A least-squares regression analysis was performed on the data. Here is some computer output from the analysis

Short Answer

Expert verified

As a result, because all conditions have been met, it is okay to proceed with the regression line.

Step by step solution

01

Given information

The given data is

02

Explanation

We must validate the following assumptions for inference about the slope of the least square regression line in this context from the previous exercise: Linear, Independent, Normal, Equal standard deviation, and Random.

Because there is no significant curvature in the residual plot, it is satisfied.

Independent: It is satisfying because the sample of $36$ high school students represents less than $10 %$ of all high school pupils.

Normal: The tallest bars in the histogram are about in the middle of the histogram, indicating that the residual has a normal distribution.

Because the vertical spread in the points of the residual plot is about the same everywhere in the plot, it is satisfied.

Because the sample is a random sample, it is satisfied.

As a result, because all conditions have been met, it is okay to proceed with the regression line.

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Most popular questions from this chapter

Brawn versus brain How is the weight of an animal’s brain related to the weight of its body? Researchers collected data on the brain weight (in grams) and body weight (in Page Number: 813 Page Number: 814 kilograms) for $96$ species of mammals. The following figure is a scatterplot of the logarithm of brain weight against the logarithm of body weight for all $96$ species. The least-squares regression line for the transformed data is

${logy}^{\land} = 1.01 + 0.72 {logx}^{\hat{\log y}} = 1.01 + 0.72 \log x$

Based on footprints and some other sketchy evidence, some people believe that a large ape-like animal, called Sasquatch or Bigfoot, lives in the Pacific Northwest. Bigfoot’s weight is estimated to be about $127$ kilograms (kg). How big do you expect Bigfoot’s brain to be?

Predicting height Using the health records of every student at a high school, the school nurse created a scatterplot relating y=height (in centimeters) to x=age (in years). After verifying that the conditions for the regression model were met, the nurse calculated the equation of the population regression line to be $μ_{y} = 105 + 4.2 x$ with $σ = 7$ cm.

a. According to the population regression line, what is the average height of $15$ -year-old students at this high school?

b. About what percent of $15$ -year-old students at this school are taller than $180$ cm?

c. If the nurse used a random sample of $50$ students from the school to calculate the regression line instead of using all the students, would the slope of the sample regression line be exactly $4.2$ ? Explain your answer.

A set of $10$ cards consists of $5$ red cards and $5$ black cards. The cards are shuffled thoroughly, and you choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and you again choose a card at random, observe its color, and replace it in the set. This is done a total of four times. Let X be the number of red cards observed in these four trials. The random variable X has which of the following probability distributions?

a. The Normal distribution with mean $2$ and standard deviation $1$

b. The binomial distribution with $n = 10$ and $p = 0.5$

c. The binomial distribution with $n = 5$ and $p = 0.5$

d. The binomial distribution with $n = 4$ and $p = 0.5$

e. The geometric distribution with $p = 0.5$

T12.11 Growth hormones are often used to increase the weight gain of chickens. In an experiment using 15 chickens, 3 chickens were randomly assigned to each of 5 different doses of growth hormone (0, 0.2, 0.4, 0.8, and 1.0 milligrams). The subsequent weight gain (in ounces) was recorded for each chicken. A researcher plots the data and finds that a linear relationship appears to hold. Here is computer output from a least-squares
regression analysis of these data. Assume that the conditions for performing inference about the slope $β_{1}$ of the true regression line are met.

a. Interpret each of the following in context:
i. The slope
ii. The $y$ intercept
iii. The standard deviation of the residuals
iv. The standard error of the slope
b. Do the data provide convincing evidence of a linear relationship between dose and weight gain? Carry out a significance test at the $α = 0.05$ level.
c. Construct and interpret a $95 %$ confidence interval for the slope parameter

Can physical activity in youth lead to mental sharpness in old age? A

2010

study investigating this question involved

9344

randomly selected, mostly white women over age

65

from four U.S. states. These women were asked about their levels of physical activity during their teenage years,

30

50

s, and later years. Those who reported being physically active as teens enjoyed the lowest level of cognitive decline-only

8.5 %

had cognitive impairment-compared with

16.7 %

of women who reported not being physically active at that time.
(a) State an appropriate pair of hypotheses that the researchers could use to test whether the proportion of women who suffered a cognitive decline was significantly smaller for women who were physically active in their youth than for women who were not physically active at that time. Be sure to define any parameters you use.
(b) Assuming the conditions for performing inference are met, what inference method would you use to test the hypotheses you identified in part (a)? Do not carry out the test.
(c) Suppose the test in part (b) shows that the proportion of women who suffered a cognitive decline was significantly smaller for women who were physically active in their youth than for women who were not physically active at that time. Can we generalize the results of this study to all women aged

65

and older? Justify your answer.
(d) We cannot conclude that being physically active as a teen causes a lower level of cognitive decline for women over

65

, due to possible confounding with other variables. Explain the concept of confounding and give an example of a potential confounding variable in this study.

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