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Chapter 10: Q. 25 (page 625)

Construct and interpret a 95% confidence interval for p1-p2 in Exercise 23. Explain what additional information the confidence interval provides.

Short Answer

Expert verified

We are 95% confident that the proportion difference is between 0.100 and0.382.

Step by step solution

01

Given Information

Given

x1=44

n1=88

x2=21

n2=81

02

Explanation

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

p^1=x1n1=4488=0.5

p^2=x2n2=2181≈0.259

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="1650451164119" p^1-p^2-zα/2·p^11-p^1n1+p^21-p^2n2=(0.5-0.259)-1.960.5(1-0.5)88+0.259(1-0.259)81≈0.100

localid="1650451174743" p^1-p^2+zα/2·p^11-p^1n1+p^21-p^2n2=(0.5-0.259)+1.960.5(1-0.5)88+0.259(1-0.259)81≈0.382

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Most popular questions from this chapter

A surprising number of young adults (ages 19to 25) still live in their parents’ homes. A random sample by the National Institutes of Health included 2253men and 2629women in this age group. The survey found that 986of the men and 923of the women lived with their parents.

(a) Construct and interpret a 99%confidence interval for the difference in population proportions (men minus women).

(b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Refer to Exercise 15.

(a) Carry out a significance test at the α=0.05level.

(b) Construct and interpret a 95%confidence interval for the difference between the population proportions. Explain how the confidence interval is consistent with the results of the test in part (a).

A driving school wants to find out which of its two instructors is more effective at preparing students to pass the state’s driver’s license exam. An incoming class of 100students is randomly assigned to two groups, each of size 50. One group is taught by Instructor A; the other is taught by Instructor B. At the end of the course, 30of Instructor A’s students and 22of Instructor B’s students pass the state exam. Do these results give convincing evidence that Instructor A is more effective?

Min Jae carried out the significance test shown below to answer this question. Unfortunately, he made some mistakes along the way. Identify as many mistakes as you can, and tell how to correct each one.

State: I want to perform a test of

H0:p1-p2=0

Ha:p1-p2>0

where p1=the proportion of Instructor A's students that passed the state exam and p2=the proportion of Instructor B's students that passed the state exam. Since no significance level was stated, I'll use σ=0.05

Plan: If conditions are met, I’ll do a two-sample ztest for comparing two proportions.

Random The data came from two random samples of 50students.

- Normal The counts of successes and failures in the two groups -30,20,22, and 28-are all at least 10.

- Independent There are at least 1000 students who take this driving school's class.

Do: From the data, p^1=2050=0.40and p^2=3050=0.60. So the pooled proportion of successes is

p^C=22+3050+50=0.52

- Test statistic

localid="1650450621864" z=(0.40-0.60)-00.52(0.48)100+0.52(0.48)100=-2.83

- p-value From Table A, localid="1650450641188" P(z≥-2.83)=1-0.0023=0.9977.

Conclude: The p-value, 0.9977, is greater than α=0.05, so we fail to reject the null hypothesis. There is no convincing evidence that Instructor A's pass rate is higher than Instructor B's.

Pat wants to compare the cost of one- and two-bedroom apartments in the area of her college campus. She collects data for a random sample of 10 advertisements of each type. The table below shows the rents (in dollars per month) for the selected apartments.

Pat wonders if two-bedroom apartments rent for significantly more, on average than one-bedroom apartments. She decides to perform a test of H0:μ1=μ2 versus Ha:μ1<μ2, where μ1 and μ2 are the true mean rents for all one-bedroom and two-bedroom aparaments, respectively, near the campus.

(a) Name the appropriate test and show that the conditions for carrying out this test are met.

(b) The appropriate test from part (a) yields a P-value of 0.058. Interpret this P-value in context.

(c) What conclusion should Pat draw at theα=0.05 significance level? Explain.

Preventing drowning Drowning in bathtubs is a major cause of death in children less than 5 years old. A random sample of parents was asked many questions related to bathtub safety. Overall, 85% of the sample said they used baby bathtubs for infants. Estimate the percent of all parents of young children who use baby bathtubs.
See all solutions

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