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Chapter 10: Q. 26 (page 625)

Construct and interpret a 95%confidence interval for p1-p2 in Exercise 24. Explain what additional information the confidence interval provides.

Short Answer

Expert verified

We are 95%confident that the proportion difference is between 0.0175 and 0.3075.

Step by step solution

01

Given Information

Given

x1=34

n1=80

x2=21

n2=80

02

Explanation

The sample proportion is the number of successes divided by the sample size:

p^1=x1n1=3480=0.425

p^2=x2n2=2180=0.2625

For confidence level 1-α=0.95, determine zα/2=z0.025using table II (look up 0.025in the table, the z-score is then the found z-score with opposite sign):

zα/2=1.96

The endpoints of the confidence interval for p1-p2are then:

localid="1650451236001" p^1-p^2-zα/2·p^11-p^1n1+p^21-p^2n2=(0.425-0.2625)-1.960.425(1-0.425)80+0.2625(1-0.2625)80≈0.0175

localid="1650451247912" p^1-p^2+zα/2·p^11-p^1n1+p^21-p^2n2=(0.425-0.2625)+1.960.425(1-0.425)80+0.2625(1-0.2625)80≈0.3075

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Most popular questions from this chapter

A large toy company introduces a lot of new toys to its product line each year. The company wants to predict the demand as measured by y, first-year sales (in millions of dollars) using x, awareness of the product (as measured by the proportion of customers who had heard of the product by the end of the second month after its introduction). A random sample of 65new products was taken, and a correlation of 0.96was computed. Which of the following is a correct interpretation of this value?

(a) Ninety-six percent of the time, the least-squares regression line accurately predicts first-year sales.

(b) About 92%of the time, the proportion of people who have heard of the product by the end of the second month will correctly predict first-year sales.

(c) About 92%of first-year sales can be explained by the proportion of people who have heard of the product by the end of the second month.

(d) The least-squares regression line relating the proportion of people who have heard of the product by the end of the second month and first-year sales will have a slope of 0.96.

(e) Ninety-two percent of the variation in first-year sales can be explained by the least-squares regression line with proportion of people who have heard of the product by the end of the second month as the explanatory variable.

Your teacher brings two bags of colored goldfish crackers to class. She tells you that Bag 1has 25%red crackers and Bag 2has 35%red crackers. Each bag contains more than 500crackers. Using a paper cup, your teacher takes an SRS of 50crackers from Bag 1 and a separate SRS of 40crackers from Bag 2. Let p1^-p2^be the difference in the sample proportions of red crackers.

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Expensive ads Consumers who think a product’s advertising is expensive often also think the product must be of high quality. Can other information undermine this effect? To find out, marketing researchers did an experiment. The subjects were 90 women from the clerical and administrative staff of a large organization. All subjects read an ad that described a fictional line of food products called “Five Chiefs.” The ad also described the major TV commercials that would soon be shown, an unusual expense for this type of product. The 45women who were randomly assigned to the control group read nothing else. The 45in the “undermine group” also read a news story headlined “No Link between Advertising Spending and New Product Quality.” All the subjects then rated the quality of Five Chefs products on a seven-point scale. The study report said, “The mean quality ratings were significantly lower in the undermine treatment (xA  4.56) than in the control treatment x¯C=5.05;t=2.64,P<0.01.

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