Statistics Confidence Intervals and Significance Test

A machine is designed to dispense at least 12 ounces of a beverage into a bottle. To test whether the machine is working properly, a random sample of 50 bottles was selected and the mean number of ounces for the 50 bottles was computed. A test of the hypotheses H0 : µ = 12 versus Ha: µ < 12 was conducted, where µ represents the population mean number of ounces of the beverage dispensed per bottle by the machine. The p-value for the test was 0.08. Which of the following is the most appropriate conclusion to draw at the significance level of α = 0.05?

• Because the p-value is greater than the significance level, there is convincing evidence that the population mean number of ounces dispensed into a bottle is less than 12 ounces.

• Because the p-value is greater than the significance level, there is not convincing evidence that the population mean number of ounces dispensed into a bottle is less than 12 ounces.

• Because the p-value is less than the significance level, there is convincing evidence that the population mean number of ounces dispensed into a bottle is 12 ounces.

• Eight percent of the bottles will be filled with less than 12 ounces.

Makers of a new pain-relieving medication claim that it relieves chronic pain faster than the current top-selling pain reliever on the market. A double-blind experiment was conducted in which 10 people who experience chronic pain were randomly selected to take either the new or the current medication. Each of the 10 people recorded the time, in minutes, from taking the medication until pain relief. After an appropriate time period, each of the 10 people took the other medication and recorded the time from taking the medication until pain relief. The medication each person took first was randomly determined, and because both medications look the same, the people in the study did not know which medication was taken first. The table below shows summary statistics for the results. Which of the following values is closest to the p-value of the appropriate t-test?

• 0.1802

• 0.3604

• 0.4230

• 0.5770

A 95 percent confidence interval of the form p̂ ±E will be used to obtain an estimate for an unknown population proportion p. If p̂ is the sample proportion and E 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

In a population of bats living in a certain region, 30 percent have a wingspan greater than 10 inches. In a random sample of 80 bats living outside of the region, 20 had a wingspan greater than 10 inches. Consider a one-sample -test to investigate whether there is evidence that the proportion of bats with a wingspan greater than 10 inches living outside the region is different from that of the bats living in the region. Which of the following is the correct test statistic?

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.

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.

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.