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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. 56 (page 818)

Tattoos: What per cent of U.S. adults have one or more tattoos? The Harris Poll conducted an online survey of $2302$ adults and found $29 %$ of respondents had at least 1 tattoo. According to the published report, “Respondents for this survey were selected from among those who have agreed to participate in Harris Interactive surveys.” Explain why it would not be appropriate to use these data to construct a $95 %$ confidence interval for the proportion of all U.S. adults who have tattoos.

Short Answer

Expert verified

The random condition was not met.

Step by step solution

01

Given information

Online survey of $2302$ adults and found $29 %$ of respondents had at least 1 tattoo.

02

Explanation

Independent, Random, and Normal are the three requirements for calculating the $95 %$ confidence interval for the proportion.

The sample was acquired using a voluntary response sample, which is not representative of the entire population, hence the random condition was not met.

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Most popular questions from this chapter

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.

How well do professional golfers putt from various distances to the hole? The scatterplot shows various distances to the hole (in feet) and the per cent of putts made at each distance for a sample of golfers.

The graphs show the results of two different transformations of the data. The first graph plots the natural logarithm of per cent made against distance. The second graph plots the natural logarithm of per cent made against the natural logarithm of distance.

a. Based on the scatterplots, would an exponential model or a power model provide a better description of the relationship between distance and per cent made? Justify your answer.

b. Here is computer output from a linear regression analysis of ln(per cent made) and distance. Give the equation of the least-squares regression line. Be sure to define any variables you use.

c. Use your model from part (b) to predict the per cent made for putts of 21 feet.

d. Here is a residual plot for the linear regression in part (b). Do you expect your prediction in part (c) to be too large, too small, or about right? Justify your answer.

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.

Do beavers benefit beetles? Researchers laid out $23$ circular plots, each $4$ meters in diameter, at random in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth so beetles prefer them. If so, more stumps should produce more beetle larvae

Here is computer output for regression analysis of these data. Construct and interpret a $99 %$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met.

role="math" localid="1654159204144" $Predictor Coef SE Coef T P Constant - 1.286 2.853 - 0.45 0.657 Stumps 11.894 1.136 10.47 0.000 S = 6.41939 R - Sq = 83.9 % R - Sq (adj) = 83.1 %$

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.

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