Proportions or Means Inference

Do the majority of YouTube channels have at least 10 videos?

• 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

Since this is a yes/no question, we need a test rather than an interval. There is only one population: YouTube channels.

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

A two-tailed hypothesis test with H₀: μ₁-μ₂ = 0 produced a p-value of 0.08.  Which of the following must be true?

• The 90% confidence interval will NOT contain 0.

• The 95% confidence interval will NOT contain 0.

• The 99% confidence interval will NOT contain 0.

• All the confidence intervals listed will contain 0.

Which of the following is an appropriate interpretation of a p-value?

• The probability of rejecting H₀, when we should have failed to reject H₀.

• The probability of failing to reject H₀, when we should have rejected H₀.

• Assuming H₀ is true, it is the probability of obtaining our sample statistic or a more extreme value.

• Assuming H₀ is NOT true, it is the probability of obtaining our sample statistic or a more extreme value.

A state educational agency was concerned that the salaries of public school teachers in one region of the state,region A, were higher than the salaries in another region of the state, region B. The agency took two independent random samples of salaries of public school teachers, one from region A and one from region B. The data are summarized in the table below. Assuming all conditions for inference are met, do the data provide convincing statistical evidence that the salaries of public school teachers in region A are, on average, greater than the salaries of public school teachers in region B?

• Yes, there is evidence at the significance level of a = 0.001.

• Yes, there is evidence at the significance level of a = 0.01 but not at a = 0.001.

• Yes, there is evidence at the significance level of a = 0.05 but not at a = 0.01.

• Yes, there is evidence at the significance level of a = 0.10 but not at a = 0.05.

• No, there is no evidence at the significance level of a = .10.

A school administrator is interested in estimating the proportion of students in the district who participate in community service activities. From a random sample of 100 students in the district, the administrator will construct a 99 percent confidence interval for the proportion of all district students who participate in community service activities. Which of the following statements must be true?

• The population proportion will be in the confidence interval.

• The probability that the confidence interval will include the population proportion is 0.99.

• The probability that the confidence interval will include the sample proportion is 0.99.

• The population proportion and the sample proportion will be equal

• The probability that the population proportion and the sample proportion will be equal is 0.99.

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

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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A representative of a car manufacturer in the United States made the following claim in a news report. "Ten years ago, only 53 percent of Americans owned American-made cars, but that figure is significantly higher today." A research group conducted a study to investigate whether the claim was true. The group found that 56 percent of a randomly selected sample of car owners in the United States owned American-made cars. A test of the appropriate hypotheses resulted in a p-value of 0.283. Assuming the conditions for inference were met, is there sufficient evidence to conclude, at the significance level of a = 0.05, that the proportion of all car owners in the United States who own American-made cars has increased from what it was ten years ago?

• Yes, because 0.56 > 0.53.

• Yes, because a reasonable interval for the proportion is 0.56 ± 0.283.

• Yes, because 0.56 - 0.53 = 0.03 and 0.03 < 0.05.

• No, because 0.283 < 0.53.

• No, because 0.283 > 0.05.