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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. 3 (page 787)

Oil and residuals Researchers examined data on the depth of small defects in the Trans-Alaska Oil Pipeline. The researchers compared the results of measurements on $100$ defects made in the field with measurements of the same defects made in the laboratory. The figure shows a residual plot for the least-squares regression line based on these data. Explain why the conditions for performing inference about the slope $β 1$ of the population regression line are not met.

Short Answer

Expert verified

Equal variance requirement has not been met.

Step by step solution

01

Given information:

We have been given that:

A residual plot for the least-squares regression line based on the data.

02

Explanation:

Conditions for performing inference:

$1 .$ Linearity

$2 .$ Independence

$3 .$ Equal Variance

$4 .$ Normality

$5 .$ Random

According to the residual plot for the least-squares regression line equal variance condition has not been met.

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Most popular questions from this chapter

In a recent poll, randomly selected New York State residents at various fast-food restaurants were asked if they supported or opposed a "fat tax" on sugared soda. Thirtyone percent said that they were in favor of such a tax and 66% were opposed. But when asked if they would support such a tax if the money raised were used to fund health care given the high incidence of obesity in the United States, $48 %$ said that they were in favor and $49 %$ were opposed.
(a) In this situation, explain how bias may have been introduced based on the way the questions were worded and suggest a way that the questions could have been worded differently in order to avoid this bias.
(b) In this situation, explain how bias may have been introduced based on the way the sample was taken and suggest a way that the sample could have been obtained in order to avoid this bias.
(c) This poll was conducted only in New York State. Suppose the pollsters wanted to ensure that estimates for the proportion of people who would support a tax on sugared soda were available for each state as well as an overall estimate for the nation as a whole. Identify a sampling method that would achieve this goal and briefly describe how the sample would be taken.

T12.3 Inference about the slope $β_{1}$ of a least-squares regression line is based on which of
the following distributions?
a. The $t$ distribution with $n - 1$ degrees of freedom
b. The standard Normal distribution
c. The chi-square distribution with $n - 1$ degrees of freedom
d. The $t$ distribution with $n - 2$ degrees of freedom
e. The Normal distribution with mean $μ$ and standard deviation $σ$ .

Prey attracts predators Here is one way in which nature regulates the size of animal populations: high population density attracts predators, which remove a higher proportion of the population than when the density of the prey is low. One study looked at kelp perch and their common predator, the kelp bass. On each of four occasions, the researcher set up four large circular pens on sandy ocean bottoms off the coast of southern California. He randomly assigned young perch to $1$ of $4$ pens so that one pen had $10$ perch, one pen had $20$ perch, one pen had $40$ perch, and the final pen had $60$ perch. Then he dropped the nets protecting the pens, allowing bass to swarm in, and counted the number of perch killed after two hours. A regression analysis was performed on the $16$ data points using x=number of perch in pen and y=proportion of perch killed. Here is a residual plot and a histogram of the residuals. Check whether the conditions for performing inference about the regression model are met.

Here is computer output from the least-squares regression analysis of the perch data.

a. Find the critical value for a $90 %$ confidence interval for the slope of the true regression line. Then calculate the confidence interval.

b. Interpret the interval from part (a).

c. Explain the meaning of “ $90 %$ confident” in this context.

Which sampling method was used in each of the following settings, in order from I to IV?

I. A student chooses to survey the first 20 students to arrive at school.

II. The name of each student in a school is written on a card, the cards are well mixed, and 10 names are drawn.

III. A state agency randomly selects 50 people from each of the state’s senatorial districts.

IV. A city council randomly selects eight city blocks and then surveys all the voting-age residents on those blocks.

a. Voluntary response, SRS, stratified, cluster

b. Convenience, SRS, stratified, cluster

c. Convenience, cluster, SRS, stratified

d. Convenience, SRS, cluster, stratified

e. Cluster, SRS, stratified, convenience

Yahtzee $(5.3, 6.3)$ In the game of Yahtzee, $5$ six-sided dice are rolled simultaneously. To get a Yahtzee, the player must get the same number on all $5$ dice.

a. Luis says that the probability of getting a Yahtzee in one roll of the dice is ${(\frac{1}{6})}^{5}$ . Explain why Luis is wrong.

b. Nassir decides to keep rolling all $5$ dice until he gets a Yahtzee. He is surprised when he still hasn’t gotten a Yahtzee after $25$ rolls. Should he be? Calculate an appropriate probability to support your answer

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