AP Statistics Inference for Proportions

A 95 percent confidence interval of the form p̂ ± ME will be used to obtain an estimate for an unknown population proportion p. If p̂ is the sample proportion and ME is the margin of error, which of the following is the smallest sample size that will guarantee a margin of error of at most 0.08?

• 25

• 100

• 175

• 250

• 625

A polling agency conducted a survey by selecting 100 random samples, each consisting of 1,200 United States citizens. The citizens in each sample were asked whether they were optimistic about the economy. For each sample, the polling agency created a 95 percent confidence interval for the proportion of all United States citizens who were optimistic about the economy. Which of the following statements is the best interpretation of the 95 percent confidence level?

• With 100 confidence intervals, we can be 95% confident that the sample proportion of citizens of the United States who are optimistic about the economy is correct.

• We would expect about 95 of the 100 confidence intervals to contain the proportion of all citizens of the United States who are optimistic about the economy.

• We would expect about 5 of the 100 confidence intervals to not contain the sample proportion of citizens of the United States who are optimistic about the economy.

• Of the 100 confidence intervals, 95 of the intervals will be identical because they were constructed from samples of the same size of 1,200.

• The probability is 0.95 that 100 confidence intervals will yield the same information about the sample

  proportion of citizens of the United States who are optimistic about the economy.

A researcher is conducting a study of charitable donations by surveying a simple random sample of households in a certain city. The researcher wants to determine whether there is convincing statistical evidence that more than 50 percent of households in the city gave a charitable donation in the past year. Let p represent the proportion of all households in the city that gave a charitable donation in the past year. Which of the following are appropriate hypotheses for the researcher?

• Ho: p = 0.5 and Ha: p > 0.5

• Ho: p = 0.5 and Ha: p ≠ 0.5

• Ho: p = 0.5 and Ha: p < 0.5

• Ho: p > 0.5 and Ha: p ≠ 0.5

• Ho: p > 0.5 and Ha: p = 0.5

The plant manager of a company that makes pillows claims that only 8 percent of the pillows made have a stitching defect. The quality control director thought that the percent might be different from 8 percent and selected a random sample of pillows to test. The director tested the hypotheses Ho: p = 0.08 versus Ha: p ≠ 0.08 at the significance level of α = 0.05. The p-value of the test was 0.03. Assuming all conditions for inference were met, which of the following is the correct conclusion?

• The p-value is less than α, and the null hypothesis is rejected. There is convincing evidence to suggest the true proportion of stitching defects is less than 0.08.

• The p-value is less than α, and the null hypothesis is rejected. There is convincing evidence to suggest the true proportion of stitching defects is not 0.08.

• The p -value is less than α, and the null hypothesis is rejected. There is convincing evidence to suggest the true proportion of stitching defects is greater than 0.08.

• The p-value is less than α, and the null hypothesis is not rejected. There is convincing evidence to suggest the true proportion of stitching defects is not 0.08.

• The p-value is less than α, and the null hypothesis is not rejected. There is not convincing evidence to suggest the true proportion of stitching defects is not 0.08.

At a large company, employees can take a course to become certified to perform certain tasks. There is an exam at the end of the course that needs to be passed for certification. The current pass rate is 0.7, but a new program is being tested to help increase the pass rate. The null hypothesis of the test is that the pass rate for the new program is 0.7. The alternative is that the pass rate for the new program is greater than 0.7. Which of the following describes a Type II error that could result from the test?

• The test does not provide convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.8.

• The test does not provide convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.7.

• The test does not provide convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.6.

• The test provides convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.8.

• The test provides convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.6.

Two voting districts, C and M, were sampled to investigate voter opinion about tax spending. From a random sample of 100 voters in District C, 22 percent responded yes to the question “Are you in favor of an increase in state spending on the arts?” An independent random sample of 100 voters in District M resulted in 26 percent responding yes to the question. A 95 percent confidence interval for the difference (p_c – p_m) was calculated as -0.04 ± 0.12. Which of the following is the best interpretation of the interval?

• We are 95% confident that the majority of all voters in the state favor an increase in state spending for the arts.

• We are 95% confident that less than half of all voters in the state favor an increase in state spending for the arts.

• We are 95% confident that the difference in the proportions of all voters in districts C and M who favor an increase in state spending for the arts is between -0.16 and 0.08.

• We are 95% confident that the difference in the sample proportions of voters in districts C and M who favor an increase in state spending for the arts is between -0.16 and 0.08.

• E. We are 95% confident that the proportion of all voters in the state who favor an increase in state spending for the arts is between -0.16 and 0.08.

_____9. Two locations of a fast-food restaurant, Location Q and Location W, were in a certain town with a large number of residents. A nutritionist investigated whether the proportion of orders that contained a salad was different at the two locations. The nutritionist obtained a random sample of orders from the Location Q restaurant and a random sample of orders from the Location W restaurant. Of the 215 Location Q orders, 27 contained a salad; of the 175 Location W orders, 14 contained a salad. Let phatC represent the combined sample proportion, and let nQ and nW represent the respective sample sizes for Locations Q and W. Have the conditions for inference for testing a difference in population proportions been met?

• A. No, the condition for independence has not been met because random samples were not selected from each location.

• B. No, the condition for independence has not been met because the sample sizes are too large when compared to the corresponding population sizes.

• C. No, the condition that the sampling distribution of the difference in sample proportions is approximately normal has not been met because nW(phatC) is not greater than 10.

• D. No, the condition that the sampling distribution of the difference in sample proportions is approximately normal has not been met because nQ(1-phatC) is not greater than 10.

• Yes, all conditions for making statistical inference have been met.

A factory manager selected a random sample of parts produced on an old assembly line and a random sample of parts produced on a new assembly line. The difference between the sample proportion of defective parts made on the old assembly line and the sample proportion of defective parts made on the new assembly line (old minus new) was 0.006. Under the assumption that all conditions for inference were met, a hypothesis test was conducted with the alternative hypothesis being the proportion of defective parts made on the old assembly line is greater than that of the new assembly line. The p-value of the test was 0.018. Which of the following is the correct interpretation of the p-value?

• If there is a difference of 0.018 in the proportions of all defective parts made on the two assembly lines, the probability of observing that difference is 0.006.

• If there is a difference of 0.006 in the proportions of all defective parts made on the two assembly lines, the probability of observing that difference is 0.018.

• If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference equal to 0.006 is 0.018.

• If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at least 0.006 is 0.018.

• If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at most 0.006 is 0.018.

A radio talk show host with a large audience is interested in the proportion p of adults in his listening area who think the drinking age should be lowered to 18. To find this out, he poses the following question to his listeners: "Do you think that the drinking age should be reduced to 18 in light of the fact that 18-year-olds are eligible for military service?" He asks listeners to go to his website and vote "Yes" if they agree the drinking age should be lowered and "No" if not. Of the 100 people who voted, 70 answered "Yes." Which of the following condition are violated?

I. Random Sample

II. 10% condition

III. Normal/Large Enough Counts

• I only

• II only

• III only

• I and II only

• I, II, and III

At a baseball game, 42 of 65 randomly selected people own an iPod. At a rock concert occurring at the same time across town, 34 of 52 randomly selected people own an iPod. A researcher wants to test the claim that the proportion of iPod owners at the two events is different. A 90% confidence interval for the difference (Game - Concert) in populations proportions is (-0.154, 0.138). Which of the following gives the correct outcome of the researcher's test of the claim?

• Because the interval includes 0, the researcher can conclude that the proportion of iPod owners at the two events is the same.

• Because the center of the interval is -0.008, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.

• Because the interval includes 0, the researcher cannot conclude that the proportion of iPod owners at the two events is different.

• Because the interval includes -0.008, the researcher cannot conclude that the proportion of iPod owners at the two events is different.

• Because the interval includes more negative than positive values, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.