An October 1-10, 2019, Gallup poll asked 1,526 U.S. adults how mu 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. The observational units of the study are U.S. adults. The variable is the amount of money U.S. adults planned to personally spend on Christmas gifts that year. I'd classify the variable as quantitative. 2. Define the parameter of interest in this study. (Hint: Do you want to estimate a mean or a proportion? For what population?) The parameter of this study is the amount of money spent by all U.S. adults for Christmas that year. PA = the amount of money spent by all U.S. adults for Christmas that year. 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 u. To con- struct 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? We know the mean, or average, ($942) and the sample size (1,526). For the mean, it would be M. For the sample mean, X. The variable would be p. What we are not told by Gallup is s, the sample standard deviation. . Explain in your own words what s r its in this context 4. Explain in your own words what s represents in this context. The "s" represents the sample standard deviation, In the example, the SD is the measure of the money spent on Christmas that year by the 1,526 adults. . 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 u. (Hint: You are 95% confident that what is between what two values?) . Suppose s was equal to $300. Use the 2SD method to approximate the 95% confidence for u. How does this interval compare to the interval from #5 (examine both the CHAPTER 3 Estimation: How Large Is the Effect? midpoint and the width of the interval)? Explain why the changes you observe make intuitive sense. The Impact of Sample Size . The poll involved 751 men and 775 women. 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. . Suppose the women reported planning to spend an average of $942 with standard deviation $150. Compare the margin of error of this interval (Hint: 2SD) to that in #5. Was your prediction correct? . In general, how does the width of a confidence interval change as the sample size increases? Why? You should have seen that a smaller sample size produced a wider confidence interval. This makes sense because a smaller sample size conveys less information and therefore leaves ncertainty about the actual value of the population mean. More directly, as we di cussed in Chapters 1 and 2, increasing the sample size red increasing the sample size reduces the variability in the sampling distribution; thus the SD of the sampling distribution will be smaller, and so the confidence interval will also be narrower. You also saw in Section 3.3 that the confidence level will impact the width of the interval well . So with quantitative data, there are three main factors that impact the width of the ce interval, the sample standard deviation (variability within the sample), the sample size (variability between samples), and the confidence level. Key Idea The width of a confidence interval for a population mean, #, decreases . as the confidence level decreases (for a fixed sample size and fixed sample standard deviation), or as the sample size incr eases (for a fixed confidence level and fixed sample standard deviation), or . as the sample standard deviation decreases (for a fixed confidence level and fixed sample size). If you wanted to answer the question "Do U.S. adults spend significantly less than $950 on average for Christmas gifts?," a confidence interval could answer that question. If we look at the confidence interval that was constructed assuming the standard deviation of spending to be $150, we get a confidence interval entirely below $950. So, yes, on average U.S. adults are spending significantly less than $950 on Christmas gifts. While this may be a statistically sig- nificant finding giving us strong evidence against the average being $950, it doesn't mean that less than $950. The upper end of the confidence interval is $949.53, 47 cents below $950. The margin of error of the interval is about $7 arguably would be spent on one additional gift. See Example 2.4 for more on this. Key Idea Especially with large sample sizes, even a fairly small difference between an observed and hypothesized value can be statistically significant. In such cases a confidence interval for the parameter can help to decide whether the difference is large. The Gallup poll did give us a bit more information; see Table 3.4.2. TABLE 3.4.2 Categorized responses to how much money respondents expected to spend on Christmas gifts 2019 (Gallup.com) $1,000 or Mean Mean $500-999 $250-499 $100-249 Under $100 No opinion Median (w/o zero) 2019 Oct 1-10 12 $500 5942 0. Based on this information, which appears to be a more reasonable guess ($150 or $300) for the sample standard deviation? Explain your reasoning. 1. Based on this information, do you think the theory-based inference method is valid for these data? (Hint: Do you think the sample data are roughly symmetric? What is the sample size?) The information provided in Table 3.4.2 is enough to tell us that the sample distribution is skewed. We can consider the 2SD and theory-based approaches valid as long as the sample size is large. Our original sample size of 1,526 respondents should be sufficient to consider these methods valid. But with such a strong skewness in spending amounts, you may want to be cautious or consider first transforming your data. We have calculated 90%, 95%, and 99% confidence intervals, and we have seen that using a higher confidence level produces a larger margin of error and therefore a wider confidence interval. But what does it mean to say that we have "95% confidence" that an Video 3.4.2 interval contains the parameter of interest? This next exploration leads you to investigate this