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Advanced High School Statistics: Third Edition

David M Diez, Mine Çetinkaya-Rundel, Leah Dorazio, Christopher D Barr

Section 5.4 Chapter exercises

Exercises Exercises

1. Twitter users and news, Part II.

A poll conducted in 2013 found that 52% of U.S. adult Twitter users get at least some news on Twitter, and the standard error for this estimate was 2.4%. Identify each of the following statements as true or false. Provide an explanation to justify each of your answers.
  1. The data provide statistically significant evidence that more than half of U.S. adult Twitter users get some news through Twitter. Use a significance level of \(\alpha = 0.01\text{.}\)
  2. Since the standard error is 2.4%, we can conclude that 97.6% of all U.S. adult Twitter users were included in the study.
  3. If we want to reduce the standard error of the estimate, we should collect less data.
  4. If we construct a 90% confidence interval for the percentage of U.S. adults Twitter users who get some news through Twitter, this confidence interval will be wider than a corresponding 99% confidence interval.

2. Chronic illness, Part II.

In 2013, the Pew Research Foundation reported that “45% of U.S. adults report that they live with one or more chronic conditions”, and the standard error for this estimate is 1.2%. Identify each of the following statements as true or false. Provide an explanation to justify each of your answers.
  1. We can say with certainty that the confidence interval from Exercise 5.2.8.1 contains the true percentage of U.S. adults who suffer from a chronic illness.
  2. If we repeated this study 1,000 times and constructed a 95% confidence interval for each study, then approximately 950 of those confidence intervals would contain the true fraction of U.S. adults who suffer from chronic illnesses.
  3. The poll provides statistically significant evidence (at the \(\alpha = 0.05\) level) that the percentage of U.S. adults who suffer from chronic illnesses is below 50%.
  4. Since the standard error is 1.2%, only 1.2% of people in the study communicated uncertainty about their answer.
Solution.
  1. False. Confidence intervals provide a range of plausible values, and sometimes the truth is missed. A 95% confidence interval “misses” about 5% of the time.
  2. True. Notice that the description focuses on the true population value.
  3. True. The 95% confidence interval is given by: \((42.6\%, 47.4\%)\text{,}\) and we can see that 50% is outside of this interval. This means that in a hypothesis test, we would reject the null hypothesis that the proportion is 0.
  4. False. The standard error describes the uncertainty in the overall estimate from natural fluctuations due to randomness, not the uncertainty corresponding to individuals’ responses.

3. Relaxing after work.

The General Social Survey asked the question: “After an average work day, about how many hours do you have to relax or pursue activities that you enjoy?” to a random sample of 1,155 Americans.
 1 
National Opinion Research Center, General Social Survey, 2018.
A 95% confidence interval for the mean number of hours spent relaxing or pursuing activities they enjoy was \((1.38, 1.92)\text{.}\)
  1. Interpret this interval in context of the data.
  2. Suppose another set of researchers reported a confidence interval with a larger margin of error based on the same sample of 1,155 Americans. How does their confidence level compare to the confidence level of the interval stated above?
  3. Suppose next year a new survey asking the same question is conducted, and this time the sample size is 2,500. Assuming that the population characteristics, with respect to how much time people spend relaxing after work, have not changed much within a year. How will the margin of error of the 95% confidence interval constructed based on data from the new survey compare to the margin of error of the interval stated above?

4. Testing for food safety.

A food safety inspector is called upon to investigate a restaurant with a few customer reports of poor sanitation practices. The food safety inspector uses a hypothesis testing framework to evaluate whether regulations are not being met. If he decides the restaurant is in gross violation, its license to serve food will be revoked.
  1. Write the hypothesis in words.
  2. What is a Type I Error in this context?
  3. What is a Type II Error in this context?
  4. Which error is more problematic for the restaurant owner? Why?
  5. Which error is more problematic for the diners? Why?
  6. As a diner, would you prefer that the food safety inspector requires strong evidence or very strong evidence of health concerns before revoking a restaurant’s license? Explain your reasoning.
Solution.
  1. \(H_{0}\text{:}\) The restaurant meets food safety and sanitation regulations. \(H_{A}\text{:}\) The restaurant does not meet food safety and sanitation regulations.
  2. The food safety inspector concludes that the restaurant does not meet food safety and sanitation regulations and shuts down the restaurant when the restaurant is actually safe.
  3. The food safety inspector concludes that the restaurant meets food safety and sanitation regulations and the restaurant stays open when the restaurant is actually not safe.
  4. A Type 1 Error may be more problematic for the restaurant owner since his restaurant gets shut down even though it meets the food safety and sanitation regulations.
  5. A Type 2 Error may be more problematic for diners since the restaurant deemed safe by the inspector is actually not.
  6. Strong evidence. Diners would rather a restaurant that meet the regulations get shut down than a restaurant that doesn’t meet the regulations not get shut down.

5. True or false.

Determine if the following statements are true or false, and explain your reasoning. If false, state how it could be corrected.
  1. If a given value (for example, the null hypothesized value of a parameter) is within a 95% confidence interval, it will also be within a 99% confidence interval.
  2. Decreasing the significance level (\(\alpha\)) will increase the probability of making a Type 1 Error.
  3. Suppose the null hypothesis is \(p = 0.5\) and we fail to reject \(H_{0}\text{.}\) Under this scenario, the true population proportion is 0.5.
  4. With large sample sizes, even small differences between the null value and the observed point estimate, a difference often called the effect size, will be identified as statistically significant.

6. Practical vs. statistical significance.

Determine whether the following statement is true or false, and explain your reasoning: “With large sample sizes, even small differences between the null value and the observed point estimate can be statistically significant.”
Solution.
True. If the sample size gets ever larger, then the standard error will become ever smaller. Eventually, when the sample size is large enough and the standard error is tiny, we can find statistically significant yet very small differences between the null value and point estimate (assuming they are not exactly equal).

7. Same observation, different sample size.

Suppose you conduct a hypothesis test based on a sample where the sample size is \(n = 50\text{,}\) and arrive at a p-value of 0.08. You then refer back to your notes and discover that you made a careless mistake, the sample size should have been \(n = 500\text{.}\) Will your p-value increase, decrease, or stay the same? Explain.
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