Section5.5Chapter exercises

Exercises5.5.1Exercises

1.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  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?

National Opinion Research Center, General Social Survey, 2018.
Solution

(a) We are 95% confident that Americans spend an average of 1.38 to 1.92 hours per day relaxing or pursuing activities they enjoy.

(b) Their confidence level must be higher as the width of the confidence interval increases as the confidence level increases.

(c) The new margin of error will be smaller, since as the sample size increases, the standard error decreases, which will decrease the margin of error.

2.Minimum wage, Part 2.

In Exercise 5.3.11.7, we learned that a Rasmussen Reports survey of 1,000 US adults found that 42% believe raising the minimum wage will help the economy. Construct a 99% confidence interval for the true proportion of US adults who believe this.

3.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

(a) $H_{0}\text{:}$ The restaurant meets food safety and sanitation regulations. $H_{A}\text{:}$ The restaurant does not meet food safety and sanitation regulations.

(b) 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.

(c) 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.

(d) 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.

(e) A Type 2 Error may be more problematic for diners since the restaurant deemed safe by the inspector is actually not.

(f) 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.

4.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.

5.Unemplyment and relationship problems.

A USA Today/Gallup poll asked a group of unemployed and underemployed Americans if they have had major problems in their relationships with their spouse or another close family member as a result of not having a job (if unemployed) or not having a full-time job (if underemployed). 27% of the 1,145 unemployed respondents and 25% of the 675 underemployed respondents said they had major problems in relationships as a result of their employment status.

1. What are the hypotheses for evaluating if the proportions of unemployed and underemployed people who had relationship problems were different?

2. The p-value for this hypothesis test is approximately 0.35. Explain what this means in context of the hypothesis test and the data.

Solution

(a) $H_{0} : p_{\text{punemp}} = p_{\text{punderemp}}\text{:}$ The proportions of unemployed and underemployed people who are having relationship problems are equal. $H_{A} : p_{\text{punemp}} \ne p_{\text{punderemp}}\text{:}$ The proportions of unemployed and underemployed people who are having relationship problems are different.

(b) If in fact the two population proportions are equal, the probability of observing at least a 2% difference between the sample proportions is approximately 0.35. Since this is a high probability we fail to reject the null hypothesis. The data do not provide convincing evidence that the proportion of of unemployed and underemployed people who are having relationship problems are different.

6.Nearsighted.

It is believed that nearsightedness affects about 8% of all children. In a random sample of 194 children, 21 are nearsighted. Conduct a hypothesis test for the following question: do these data provide evidence that the 8% value is inaccurate?

7.Nutrition labels.

The nutrition label on a bag of potato chips says that a one ounce (28 gram) serving of potato chips has 130 calories and contains ten grams of fat, with three grams of saturated fat. A random sample of 35 bags yielded a confidence interval for the number of calories per bag of 128.2 to 139.8 calories. Is there evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips?

Solution

Because 130 is inside the confidence interval, we do not have convincing evidence that the true average is any different than what the nutrition label suggests.

8.CLT for proportions.

Define the term “sampling distribution” of the sample proportion, and describe how the shape, center, and spread of the sampling distribution change as the sample size increases when $p = 0.1\text{.}$

9.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).

10.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.

11.Gender pay gap in medicine.

A study examined the average pay for men and women entering the workforce as doctors for 21 different positions. 2

1. If each gender was equally paid, then we would expect about half of those positions to have men paid more than women and women would be paid more than men in the other half of positions. Write appropriate hypotheses to test this scenario.

2. Men were, on average, paid more in 19 of those 21 positions. Complete a hypothesis test using your hypotheses from part (a).

Solution

(a) In effect, we're checking whether men are paid more than women (or vice-versa), and we'd expect these outcomes with either chance under the null hypothesis:

\begin{align*} H_{0}: p=0.5 \amp \amp H_{A} p \ne 0.5 \end{align*}

We'll use p to represent the fraction of cases where men are paid more than women.

Below is the completion of the hypothesis test.

• There isn't a good way to check independence here since the jobs are not a simple random sample. However, independence doesn't seem unreasonable, since the individuals in each job are different from each other. The success-failure condition is met since we check it using the null proportion: $p_{0}n = (1-p_{0})n = 10.5$ is greater than 10.

• We can compute the sample proportion, SE, and test statistic:

\begin{gather*} \hat{p}=19/21 = 0.905\\ SE=\sqrt{\frac{0.5 \times (1-0.5)}{21}}=0.109\\ Z=\frac{0.905-0.5}{0.109}=3.72 \end{gather*}

The test statistic $Z$ corresponds to an upper tail area of about 0.0001, so the p-value is 2 times this value: 0.0002.

• Because the p-value is smaller than 0.05, we reject the notion that all these gender pay disparities are due to chance. Because we observe that men are paid more in a higher proportion of cases and we have rejected $H_{0}\text{,}$ we can conclude that men are being paid higher amounts in ways not explainable by chance alone.