Exam 12: Inferences for Population Proportions

The number of successes and the sample size are given for a simple random sample from a population. Decide whetherusing the one-proportion z-test is appropriate. - $x = 7 , n = 270 , \mathrm { H } _ { 0 } : \mathrm { p } = 0.04 , \mathrm { H } _ { \mathrm { a } } : \mathrm { p } \neq 0.04$

A researcher is interested in estimating the proportion of voters who favor a tax on e-commerce. Based on a sample of 250 people, she obtains the following 99% confidence interval for the Population proportion p: 0.113 < p < 0.171 Which of the statements below is a valid interpretation of this confidence interval?

In a poll of 2490 voters in a certain city, 61% said that they backed a bill which would limit growth and development in their city. The margin of error in the poll was reported as 2 percentage points (with a 95% degree of confidence). Which statement is correct?

Suppose that you wish to estimate a population proportion and want to determine a sample size that will ensure a given margin of error for a $95 \%$ confidence interval. Suppose further that an educated guess of $0.3$ is used for $\hat { p }$ when determining the sample size. What values of the observed value of $\hat { p }$ will yield a larger margin of error than the one specified?

The number of successes and the sample size are given for a simple random sample from a population. Use theone-proportion plus-four z-interval procedure to find the required confidence interval. -n = 188, x = 157; 95% level

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. -A random sampling of sixty pitchers from the National League and fifty-two pitchers from the American League showed that 19 National and 6 American League pitchers had E.R.A's below 3.5. At the 1% significance level, do the data provide sufficient evidence to conclude that the proportion of National League pitchers with E.R.A's below 3.5 differs from the proportion of American League pitchers with E.R.A's below 3.5 ?

Give an example of a situation in which you might wish to use the two-proportions z-test. Identify the two populations in your example and the attribute of interest. State what you would wish to determine by performing the hypothesis test and write the hypotheses.

Use the two-proportions z-interval procedure to obtain the required confidence interval for the difference between twopopulation proportions. Assume that independent simple random samples have been selected from the two populations. -In a random sample of 45 Democrats from one city, 10 approved of the mayor's performance. In a random sample of 49 Republicans from the city, 24 approved of the mayor's performance. Find a 90% confidence interval for the difference between the proportions of Democrats and Republicans Who approve of the mayor's performance.

The numbers of successes and the sample sizes are given for independent simple random samples from two populations.Use the two-proportions z-interval procedure to obtain the specified confidence interval. - $\mathrm { x } _ { 1 } = 32 , \mathrm { n } _ { 1 } = 80 , \mathrm { x } _ { 2 } = 17 , \mathrm { n } _ { 2 } = 20.99 \%$ confidence interval

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 random sample of 500 people aged 20-24, 22% were smokers. In a random sample of 450 people aged 25-29, 14% were smokers. Do the data provide sufficient evidence to conclude that the proportion of smokers in the 20-24 age group is different from the proportion of smokers in the 25-29 age group? Use a significance level of 0.01.

Use the two-proportions z-interval procedure to obtain the required confidence interval for the difference between twopopulation proportions. Assume that independent simple random samples have been selected from the two populations. -A survey found that 28 of 69 randomly selected women and 41 of 78 randomly selected men follow a regular exercise program. Find a 95% confidence interval for the difference between the Proportions of women and men who follow a regular exercise program.

The number of successes and the sample size are given for a simple random sample from a population. Use theone-proportion z-interval procedure to find the required confidence interval. -n = 138, x = 86; 90% level

Suppose that you wish to estimate a population proportion and want to determine a sample size that will ensure a given margin of error for a $95 \%$ confidence interval. Suppose further that an educated guess is not available for $\hat { \mathrm { p } }$ and that $0.25$ is used for the value of $\hat { \mathrm { p } } ( 1 - \hat { \mathrm { p } } )$ when determining the sample size. True or false, if the observed value of $\hat { \mathrm { p } }$ is not equal to $0.5$ , then the sample size will be larger than necessary?

In a poll of 2500 voters in a certain city, 66% said that they backed a bill which would limit growth and development in their city. The margin of error in the poll was reported as 2 percentage points (with a 95% degree of confidence). Which statement is correct?

Use the one-proportion z-test to perform the required hypothesis test. Use the P-value approach. -An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Littleton. A random sample of 233 fathers from Littleton yielded 96 who did not help with child care. Do the data provide sufficient evidence to conclude that in Littleton the percentage of fathers who take no responsibility for child care is higher than 34%? Use a 0.05 significance level.

What is the null hypothesis for the two-proportions z-test? What assumptions are required for this test?

Find the P-value for the indicated hypothesis test. -A medical school claims that more than 28% of its students plan to go into general practice. It is found that among a random sample of 130 of the school's students, 39% of them plan to go into General practice. Find the P-value for a test of the school's claim.

Use the one-proportion z-test to perform the required hypothesis test. Use the P-value approach. -In a sample of 165 children selected randomly from one town, it is found that 30 of them suffer from asthma. At the 0.05 significance level, do the data provide sufficient evidence to conclude that the percentage of all children in the town who suffer from asthma is different from 11%?

Use the one-proportion z-test to perform the required hypothesis test. Use the critical-value approach. -In a sample of 90 adults selected randomly from one town, it is found that 6 of them have been exposed to a particular strain of the flu. At the 0.01 significance level, test whether the proportion of all adults in the town that have been exposed to this strain of the flu differs from the nationwide percentage of 8%.

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. -In a random sample of 360 women, 65% favored stricter gun control laws. In a random sample of 220 men, 60% favored stricter gun control laws. At the 0.05 significance level, do the data provide sufficient evidence to conclude that the proportion of women favoring stricter gun control is higher than the proportion of men favoring stricter gun control?