Exam 12: Inferences for Population Proportions

Use the one-proportion z-test to perform the required hypothesis test. Use the critical-value approach. -A newspaper article citing the results of a poll states: "In theory, the results of such a poll, in 99 cases out of 100 should differ by no more than 2 percentage points in either direction from what Would have been obtained by interviewing all voters in the United States." Find the sample size Suggested by this statement.

In the context of a two-proportions z-test, explain what each of the following symbols represents: $\mathrm { x } _ { 1 } , \mathrm { n } _ { 1 } , \mathrm { p } 1 , \mathrm { p } 1 , \mathrm { p } _ { \mathrm { p } }$ .

A population consists of four men and one woman. The first names of the men are Adam, Bernard, Charlie, and Daniel. The first name of the woman is Elena. Suppose that the specified attribute is "male". a. Determine the population proportion, p. b. Complete the table below. The first column shows the possible samples of size 3, the second column gives the number of successes - the number of males obtained - for each sample, and the third column shows the sample proportion. Sample Number of males Sample proportion A, B, C 3 1 A, B, D 3 1 A, B, E 2 2/3 A, C, D A, C, E A, D, E B, C, D B, C, E B, D, E C, D, E c. Use the third column of the table to obtain the mean of the variable $\hat { p }$ . Is this mean smaller, larger, or the same as your answer in part a?. Why is this the case?

Use the one-proportion z-test to perform the required hypothesis test. Use the P-value approach. -A supplier of 3.5" disks claims that no more than 1% of the disks are defective. In a random sample of 600 disks, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. At the 0.01 level of significance, do the data provide sufficient evidence that the percentage of defects exceeds 1%?

Use the one-proportion z-test to perform the required hypothesis test. Use the P-value approach. -In a clinical study of an allergy drug, 108 of the 201 subjects reported experiencing significant relief from their symptoms. At the 0.01 significance level, do the data provide sufficient evidence to conclude that more than half of those using the drug experience relief?

The number of successes and the sample size are given for a simple random sample from a population. Determine thesample proportion, $\hat{ p. }$ -x = 16, n = 25

Suppose the proportion of sophomores at a particular college who purchased used textbooks in the past year is ps and the proportion of freshmen at the college who purchased used textbooks in the past year is pf. A study found a 90% confidence interval for $\mathrm { p } _ { \mathrm { s } } = \mathrm { pf } _ { \mathrm { f } } \text { to be } ( 0.239,0.429 )$ Does this interval suggest that sophomores are more likely than freshmen to buy used textbooks? Explain.

Use the two-proportions z-test to perform the required hypothesis test. Assume that independent simple randomsamples have been selected from the two populations. Use the P-value approach. -A marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 recognized the product while 196 out of 614 Californians recognized the product. At the 5% significance level, do the data provide sufficient evidence to conclude that the recognition rate in New York differs from the recognition rate in California?

Use the two-proportions z-test to perform the required hypothesis test. Assume that independent simple randomsamples have been selected from the two populations. Use the critical-value approach. -In a survey, 84 of 185 college graduates said that they were satisfied in their work, while 76 of 180 adults without a college education said that they were satisfied with their work. At the 5% significance level, do the data provide sufficient evidence to conclude that college graduates are more likely to be satisfied with their work than those without a college education?

A researcher wishes to determine whether the proportion of American women who smoke differs from the proportion of American men who smoke. He wants to test the hypothesis $\mathrm { H } _ { 0 } : \mathrm { p } _1 = \mathrm { p } _ { 2 }$ where p1 represents the proportion of American women who smoke and p2 represents the proportion of American men who smoke. He randomly selects 100 married couples. Among the 100 women in the sample are 21 smokers. Among the 100 men are 29 smokers. Are the assumptions for a two-proportions z-test satisfied? If not, which assumption is violated and why?

A researcher wishes to test whether the rate of defectives among computers from manufacturer A differs from the rate of defectives among computers from manufacturer B. She selects two independent random samples and finds that 1.5% of 200 computers from manufacturer A are defective and 3.5% of 400 computers from manufacturer B are defective. Are the conditions and assumptions for inference with the two-proportion z-test satisfied? Explain your answer.

Determine whether the following statement regarding a hypothesis test for two population proportions. However small the difference between two population proportions, for sufficiently large sample sizes, the null hypothesis of equal population proportions is likely to be rejected.

The numbers of successes and the sample sizes are given for independent simple random samples from two populations.Use the two-proportions plus-four z-interval procedure to obtain the specified confidence interval. - $\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 60 , \mathrm { x } _ { 2 } = 29 , \mathrm { n } _ { 2 } = 77 ; 95 \%$ confidence interval

Use the one-proportion z-test to perform the required hypothesis test. Use the P-value approach. -A research group claims that less than 28% of students at one medical school plan to go into general practice. It is found that among a random sample of 120 of the school's students, 20% of them plan to go into general practice. At the 0.10 significance level, do the data provide sufficient evidence to conclude that the percentage of all students at this school who plan to go into general practice is less than 28%?

Use the one-proportion z-test to perform the required hypothesis test. Use the critical-value approach. -An airline's public relations department says that the airline rarely loses passengers' luggage. It further claims that on those occasions when luggage is lost, 92% is recovered and delivered to its owner within 24 hours. A consumer group who surveyed a large number of air travelers found that only 145 out of 165 people who lost luggage on that airline were reunited with the missing items by the next day. At the 5% level of significance, do the data provide sufficient evidence to conclude that the proportion of times that luggage is returned within 24 hours is less than 0.92?

Use the one-proportion z-test to perform the required hypothesis test. Use the P-value approach. -A poll of 1000 adult Americans reveals that 48% of the voters surveyed prefer the Democratic candidate for the presidency. At the 0.05 level of significance, do the data provide sufficient evidence to conclude that the percentage of voters who prefer the Democrat is less than 50%?

The college daily reported: "450 students living in university housing were polled. 270 said that they were satisfied with their living conditions. Based on this survey we conclude that 60% of Students living in dormitories are satisfied. The margin of error of the study is $\pm 5$ percentage Points (with a 95% degree of confidence). Which statement is correct?

The numbers of successes and the sample sizes are given for independent simple random samples from two populations.Use the two-proportions plus-four z-interval procedure to obtain the specified confidence interval. - $\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 38 , \mathrm { x } _ { 2 } = 17 , \mathrm { n } _ { 2 } = 38 ; 80 \%$ confidence interval

The number of successes and the sample size are given for a simple random sample from a population. Decide whetherusing the one-proportion z-interval procedure is appropriate. -x = 2, n = 40

$\text { Find the required sample size without making a guess for the observed value of } \hat { p } \text {. }$ -A researcher wishes to estimate the proportion of all drivers who exceed the speed limit on a certain stretch of road where accidents frequently happen. Obtain a sample size that will ensure a Margin of error of at most 0.021 for a 98% confidence interval.