Preparing for the next class
In the next activity, you will need to be able to identify the population of interest in a research study, use information about a sample to assess whether it is reasonable to think a population is approximately normal in a given context, and calculate and interpret the standard error of the sample mean.
When you are interested in estimating a population mean, you usually start with data from a sample from the population of interest. To estimate the population mean, you start by calculating the sample mean. The sample mean can then be used to construct a confidence interval for the population mean, in the same way that a sample proportion is used to construct a confidence interval estimate of a population proportion.
The confidence interval that you will see in the next activity is based on what we know about the behavior of the sample mean. The following properties of the sampling distribution of the sample mean were introduced in In-Class Activity 12.A.
| Sampling Distribution of the Sample Mean
When taking many random samples of size [latex]n[/latex] from a population distribution with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex]: The mean of the distribution of the sample means is [latex]\mu[/latex]. The standard deviation of the distribution of the sample means is [latex]\frac{\sigma}{\sqrt{n}}[/latex]. If the population distribution is normal or if the sample size is large [latex](n ≥ 30)[/latex], the distribution of the sample means follows an approximate normal distribution. |
Keep these properties in mind as you answer the questions in this assignment.
Question 1
1) The General Social Survey (GSS) collects data from a representative sample of adults in the United States on a number of attitudes and behaviors:
https://gss.norc.org/About-The-GSS.
One of the questions asked as part of the survey is how many hours are spent watching TV on a typical day. The dataset we will be using consists of responses from a sample of 1,555 adults from the 2018 survey.
a) If you plan to use the sample data on TV time from the GSS to estimate a population mean, what is the population of interest?
Hint: What group was sampled for the survey?
b) What is the sample size?
Hint: How many people were surveyed?
c) Here is a histogram of the sample TV time data. Is it reasonable to think that the distribution of the TV time is approximately normal? Explain.
*missing image*
d) Will the sampling distribution of the sample mean be approximately normal? Explain.
Hint: Review the properties of the sampling distribution of the sample mean at the beginning of this assignment.
e) The standard deviation of the sampling distribution of the sample mean is [latex]\frac{\sigma}{\sqrt{n}}[/latex]. This standard deviation can be estimated by the standard error [latex]\frac{s}{\sqrt{n}}[/latex], where [latex]s[/latex] is the sample standard deviation. For the TV time dataset, [latex]s=2.837[/latex]. What is the value of the standard error? Round your answer to 5 decimal places.
Hint: Substitute [latex]s[/latex] and [latex]n[/latex] into the formula for the standard error.
f) The standard error is very small, indicating that there is not much variability in the sample means from one sample of 1,555 adults to another sample of 1,555 adults. Why do you think that the standard error is so small?
Hint: Look at the formula for standard error.
g) Do you think that the sample mean from this GSS sample would be close to the actual value of the mean TV time for the whole population? Explain.
Hint: Look at the properties of the sampling distribution of the sample mean and think about what the standard error tells you.
Question 2
2) Researchers in New York carried out a study to investigate how many calories are consumed when people eat lunch at fast-food restaurants.[1] They asked people eating lunch at different locations of McDonald’s, Burger King, and Wendy’s if they would give them their receipts after they had ordered, and then they used the receipts to see what had been ordered to determine the number of calories in the meals. A total of 3,857 meals were analyzed in the study, and the researchers believed that this sample was representative of lunch meals eaten at fast-food restaurants.
a) If you plan to use the sample data on calorie content from this study to estimate a population mean, what is the population of interest?
Hint: What was the goal of the study? What did the researchers hope to learn?
b) What is the sample size?
Hint: How many meals were analyzed in the study?
c) The mean calorie content for the sample was 857 calories and the sample standard deviation was 677 calories. Thinking about this mean and standard deviation and the fact that calorie content can’t be negative, explain why it is not reasonable to think that the distribution of the calorie content is approximately normal?
Hint: Think about what the Empirical Rule tells you about the distribution of values for distributions that are symmetric and mound shaped or about what you know about normal distributions.
d) Will the sampling distribution of the sample mean be approximately normal? Explain.
Hint: Review the properties of the sampling distribution of the sample mean at the beginning of this assignment.
e) The standard deviation of the sampling distribution of the sample mean is [latex]\frac{\sigma}{\sqrt{n}}[/latex]. This standard deviation can be estimated by the standard error [latex]\frac{s}{\sqrt{n}}[/latex], where [latex]s[/latex] is the sample standard deviation. For the calorie content dataset, [latex]s=677[/latex]. What is the value of the standard error? Round your answer to 4 decimal places.
Hint: Substitute [latex]s[/latex] and [latex]n[/latex] into the formula for the standard error.
f) The standard error is very small (e.g., 1 potato chip), indicating that there is not much variability in the sample means from one sample of 3,857 lunch meals to another sample of 3,857. Why do you think that the standard error is so small?
Hint: Look at the formula for standard error.
g) Do you think that the sample mean from this sample would be close to the actual value of the mean calorie content for the population of fast-food lunch meals? Explain.
Hint: Look at the properties of the sampling distribution of the sample mean and think about what the standard error tells you.
- Dumanovsky, T., Nonas, C. A., Huang, C. Y., Silver, L. D., & Bassett, M. T. (2009, July). What people buy from fast-food restaurants: Caloric content and menu item selection, New York City 2007. Obesity 17(7), 1369–1374. https://onlinelibrary.wiley.com/doi/full/10.1038/oby.2009.90 ↵
