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The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset Vaia Explanations$ Math

4th Edition · 809 pages

ISBN: 9781319113339

Chapter 12: Q. 2 (page 759)

SAT Math scores In Chapter 3, we examined data on the percent of high school graduates in each state who took the SAT and the state's mean SAT Math score in a recent year. The figure below shows a residual plot for the least-squares regression line based on these data. Are the conditions for performing inference about the slope $β$ of the population regression line met? Justify your answer.

Short Answer

Expert verified

The linear requirement has not been met.

Step by step solution

01

Given Information

Need to find which requirement has not been met.

02

Explanation

There are five conditions for regression inference are: Linear, Independent, Normal, Equal variance, Random

The Linear requirement has not been met, because the residual plot contains strong curvature and thus the scatterplot is not roughly linear.

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Most popular questions from this chapter

In the casting of metal parts, molten metal flows through a “gate” into a die that shapes the part. The gate velocity (the speed at which metal is forced through the gate) plays a critical role in die casting. A firm that casts cylindrical aluminium pistons examined a random sample of $12$ pistons formed from the same alloy of metal. What is the relationship between the cylinder wall thickness (inches) and the gate velocity (feet per second) chosen by the skilled workers who do the casting? If there is a clear pattern, it can be used to direct new workers or to automate the process. A scatterplot of the data is shown below

A least-squares regression analysis was performed on the data. Some computer output and a residual plot are shown below. A Normal probability plot of the residuals (not shown) is roughly linear.

Do these data provide convincing evidence of a straight-line relationship between thickness and gate velocity in the population of pistons formed from this alloy of metal? Carry out an appropriate significance test at the $α = 0.05$ level.

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, $30 s$ , $50 s$ , 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 lower 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 (b)? 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 lower 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.

Which of the following would provide evidence that a power law model of the form $y = a x^{b}$ where $b \neq 0$ and $b \neq 1$ , describes the relationship between a response variable $y$ and an explanatory variable $x .$

(a) A scatterplot of $y$ versus $x$ looks approximately linear.

(b) A scatterplot of ln $y$ versus $x$ looks approximately linear.

(c) A scatterplot of $y$ versus ln $x$ looks approximately linear.

(d) A scatterplot of ln $y$ versus ln $x$ looks approximately linear.

(e) None of these.

Suppose a name-brand drug has been deemed effective for reducing hypertension (high blood pressure). The developing company gets to keep a patent on the drug for a specific period of time before other companies can develop a generic form of the drug. Suppose the patent period is about to expire, and another company produces a generic version of this drug. The Food and Drug Administration (FDA) wants to know whether the generic drug is at least as effective as the name-brand drug in reducing blood pressure.

The following hypotheses will be used:

$H_{0} : μ_{g} = μ_{n}$ vs $H_{a} : μ_{g} < μ_{n}$

where

$μ_{g} =$ true mean reduction in blood pressure using the generic drug

$μ_{n} =$ true mean reduction in blood pressure using the name-brand drug. In the context of this situation, which of the following describes a Type I error?

(a) The FDA finds sufficient evidence that the generic drug does not reduce blood pressure as much as the namebrand drug when, in fact, it does not.

(b) The FDA finds sufficient evidence that the generic drug does not reduce blood pressure as much as the namebrand drug when, in fact, it does.

(c) The FDA finds sufficient evidence that the generic drug does reduce blood pressure as much as the namebrand drug when, in fact, it does not.

(d) The FDA finds sufficient evidence that the generic drug does reduce blood pressure as much as the namebrand drug when, in fact, it does.

(e) The FDA does not find sufficient evidence that the generic drug is as effective in reducing blood pressure as the name-brand drug when, in fact, it is.

Some high school physics students dropped a ball and measured the distance fallen (in centimeters) at various times (in seconds) after its release. If you have studied physics, then you probably know that the theoretical relationship between the variables is distance $= 490 {(t i m e)}^{2}$ . A scatterplot of the students’ data showed a clear curved pattern. A scatterplot of which of the following should have a roughly linear pattern?

(a) [time, ln(distance)]

(b) [ln(time), distance]

(c) [ln(time), ln(distance)]

(d) [ln(distance), time]

(e) [distance, ln(time)]

See all solutions

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