Inference for the Difference in Proportions

A credit card company would like to compare the proportion of people who have poor credit to the proportion of people who think they have poor credit. Which of the following is most appropriate?

• Proportion Z-Test

• Proportion Z-Interval

• 2-Proportion Z-Test

• 2-Proportion Z-Interval

Which of the following is true pertaining to a 2-proportion z-interval?

• There are no restrictions on sample size 

• This interval can be found even if the samples are not random

• The population should be large, relative to each of the sample sizes

• So long as all conditions are met for one sample, you do not need to check conditions for the other sample

What additional condition must be checked for a 2-proportion test or interval compared to a 1-proportion test or interval?

• You must make sure that the samples are random

• You must make sure that the two samples are independent from one another

• You must make sure that the sum of the samples' sizes are smaller than 10% of the whole population

• The conditions are exactly the same

In a past General Social Survey, a random sample of men and women answered the question “Are you a member of any sports clubs?” Based on the sample data, 95% confidence intervals for the population proportion who would answer “yes” are .13 to .19 for women and .247 to .33 for men. Based on these results, you can reasonably conclude that 

• At least 25% of American men and American women belong to sports clubs.

• At least 16% of American women belong to sports clubs. 

• There is a difference between the proportions of American men and American women who belong to sports clubs.

• There is no conclusive evidence of a gender difference in the proportion belonging to sports clubs

Two large containers, X and Y, contain many colored beads. From a random sample of beads taken from container X, the proportion of blue beads in the sample was recorded as phat_x=0.35. From a random sample of beads taken from container Y, the proportion of blue beads in the sample was recorded as phat_y=0.39. Assuming all conditions for inference are met, which of the following procedures is the most appropriate for estimating the difference between the proportions of all blue beads in the containers?

• A two-sample z-interval for a difference in population proportions

• A two-sample z-interval for a difference in sample proportions

• A one-sample z-interval for a population proportion

• A one-sample z-interval for a sample proportion

• A one-sample z-interval for a difference in population proportions

Answer explanation

A two-sample z-interval is appropriate for estimating the difference between two population proportions.

Should there be more restrictions on handguns? In a pre-Columbine survey, 255 out of 1,020 adults answered in the affirmative; in a 2000 post-Columbine survey, 352 out of 1,100 answered affirmatively. Establish a 90 percent confidence interval estimate of the difference between the proportions of adults in 1995 and 2000 who support more restrictions on handguns.

As part of a national sleep study, a random sample of adults was selected and surveyed about their physical activity and the number of hours they sleep each night. Of the 183 adults who exercised regularly (exercisers), 59 percent reported sleeping at least seven hours at night. Of the 88 adults who did not exercise regularly (nonexercisers), 52 percent reported sleeping at least seven hours at night. Which of the following is the most appropriate standard error for a confidence interval for the difference in proportions of adults who sleep at least seven hours at night among exercisers and nonexercisers?

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Are people who have pets more likely to be left-handed than people who don't have pets?

• one-sample z test for a proportion

• one-sample z interval for a proportion

• two-sample z test for the difference of proportions

• two-sample z intervalfor the difference of proportions

Answer explanation

This is a yes/no question, so a test is required rather than an interval. There are two separate populations: people who have pets vs people who don't, so a two-sample test is needed.

How much more likely are teenagers to use TikTok compared to people over the age of 20?

• one-sample z test for a proportion

• one-sample z interval for a proportion

• two-sample z test for the difference of proportions

• two-sample z interval for the difference of proportions

Answer explanation

The question asks for the size of the difference, so an interval is needed rather than a test. There are two separate populations: teenagers and people over the age of 20.

If p₁ - p₂ = 0, is in the 90% confidence interval, what can we conclude at α = 0.10?

• Reject H₀, there is sufficient evidence to conclude H_a

• Reject H₀, there is NOT sufficient evidence to conclude H_a

• Fail to reject H₀, there is sufficient evidence to conclude H_a

• Fail to reject H₀, there is NOT sufficient evidence to conclude H_a