From October 1-10, 2019, Gallup poll asked 1,526 U.S. adults how much they planned to personally spend on Christmas gifts this year. The report cited an average of $942.

1. Identify the observational units and variable in this study. Classify the variable as quantitative or categorical.

2. Define the parameter of interest in this study. (Hint: Do you want to estimate a mean or a proportion? For what population?)

Because our variable of interest (dollar amount intending to spend) is quantitative, the parameter of interest here is the population mean for all adult Americans; call this . To construct a confidence interval for , we need to know the sample size, the sample mean, and the sample standard deviation.

3. Which two of these three quantities do you know? What symbols can be used to represent them?

What we are not told by Gallup is s, the sample standard deviation

4. Explain in your own words what s represents in this context.

5. Suppose s was equal to $150. Use the 2SD method (Hint: Reminder to consider the SD of the sample mean) to approximate and interpret a 95% confidence interval for . (Hint: You are 95% confident that what is between what two values?)

6. Suppose s was equal to $300. Use the 2SD method to approximate the 95% confidence interval for . How does this interval compare to the interval from #5 (examine both the midpoint and the width of the interval)? Explain why the changes you observe make intuitive sense.

7. The poll involved 751 people from Kansas and 775 people from Nebraska. Suppose we looked at only one of these subgroups. How do you expect the width of the intervals to compare to the interval based on the total sample (wider, narrower, or the same width as before)? Explain briefly.

The2011 Statistical Abstract of the United Statesincludes a table reporting that a national survey in the fall of 2009 found that 18.2% of American adults played a board game in the past year. We can use this sample result to estimate the proportion ofalladult Americans who had played a board game in the past year.

a. What additional information, do you need to determine a 95% confidence interval for the probability that a randomly selected American adult had played a board game in the past year?

b. Consider three possible sample sizes for this study: 250, 1,000, and 4,000. Without performing any calculations, how do you expect three confidence intervals based on these sample sizes to compare to each other? Explain your reasoning.

c. Carry out the simulation at each sample size to get the SD and use the 2SD method to approximate the margin of error with each of these three sample sizes. Then produce the 95% confidence interval for each of these three sample sizes.

d. By how many times must the sample size increase to cut the margin of error in half?