There are many situations where you might be interested in estimating a population mean. For example, you might be interested in collecting data from a random sample of students who graduated from a two-year college in 2020 to learn about student loans. If you asked each student in the sample the amount of their student loan debt, you could then use the data to estimate the mean student loan debt for two-year college graduates.
Question 1
1) What are examples of other situations where you might want to estimate a population mean?
In Lesson 10, you constructed confidence interval estimates for a population proportion and a difference in proportions when certain assumptions/conditions were met. The form of those confidence intervals was
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estimate ± margin of error
where the margin of error was calculated by multiplying the standard error of the estimate by a z-critical value corresponding to the desired confidence level.
When the population parameter that you are interested in estimating is a population mean, the confidence interval has the same form. The estimate used to construct the interval is the sample mean, [latex]\bar{x}[/latex], and the standard error used is the standard error of the sample mean, [latex]\frac{s}{\sqrt{n}}[/latex].
The margin of error is calculated a little differently—instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate t Distribution. This is not surprising if you think back to your work with the standardized t-statistic in In-Class Activity 12.B.
The formula for a confidence interval for a population mean is
[latex]\bar{x}\pm(t-critical\;value)\frac{s}{\sqrt{n}}[/latex]
The t-critical value (t-score in the data analysis tool) in the confidence interval will depend on the sample size (degrees of freedom for the t Distribution [latex]=n-1[/latex]) and the confidence level. This interval is often called a one-sample t interval.
While you will use technology to do the calculations, you can see from this confidence interval formula that you are just taking the sample mean and forming an interval around it by subtracting and adding the margin of error to get an interval of plausible values for the population mean.
Question 2
2) 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) Describe the population mean that these data could be used to estimate. Be sure your description includes the population of interest.
b) Go to the DCMP Inference for a Population Mean tool at
https://dcmathpathways.shinyapps.io/Inference_mean/. You will use this tool to calculate confidence intervals for a population mean. Under the
Confidence and Significance Tests tab:
- Select the TV Hours dataset.
- For Type of Inference, select “Confidence Interval.”
- Use the slider for the confidence level to select a 90% confidence level.
Based on the output, what is the 90% confidence interval for the mean number of hours spent watching TV on a typical day for adults in the United States?
c) Think about how you have previously interpreted confidence intervals for proportions. How would you interpret the confidence interval you just calculated?
d) Which of the following is a correct interpretation of the confidence level of 90%?
(a) The probability that the actual value of the population mean is between 2.820 and 3.057 hours is 0.90.
(b) 90% of U.S. adults watch between 2.820 and 3.057 hours of TV on a typical day.
(c) If this method was used to construct a confidence interval for the mean for many different samples from the population, about 90% of the intervals would contain the actual population mean.
(d) The mean number of hours that U.S. adults spend watching TV on a typical day is guaranteed to be in the interval from 2.820 to 3.057.
e) If you were to use this sample to calculate a 95% confidence interval rather than a 90% confidence interval, how would the width of the two intervals compare?
(a) The width of the two intervals would be the same.
(b) The 95% confidence interval will be wider than the 90% confidence interval.
(c) The 95% confidence interval will be narrower than the 90% confidence interval.
f) Use the tool to compute the 95% confidence interval. Is it consistent with your answer to Part E? Explain.
As was the case with previous inference methods, there are a few assumptions/conditions that you should check before using the one-sample t interval. Two important ones are:
- The sample is a random sample from the population of interest or it is reasonable to regard the sample as if it were a random sample. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in a sample that is representative of the population.
- The population distribution of the variable that was measured is approximately normal, or the sample size is large. Usually, a sample of size 30 or more is considered to be “large.” If the sample size is less than 30, you should look at a plot of the data (a dotplot, a boxplot, or, if the sample size isn’t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.
For the TV hours example, the sample size is large and the sample was selected to be representative of adults in the United States, so the one-sample t confidence interval is an appropriate way to estimate the population mean.
Question 3
3) 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.
The mean calorie content for the sample was 857 calories and the sample standard deviation was 677 calories.
a) Is the one-sample t interval an appropriate way to estimate the mean calorie content of fast-food lunches? Recall that in the preview assignment, you decided that the population distribution of the calorie content for fast food lunches was not normal. Are the assumptions for the one-sample t interval reasonable?
b) Use the tool to calculate a 95% confidence interval for the mean calorie content of fast-food lunches. Go to the DCMP Inference for a Population Mean tool at https://dcmathpathways.shinyapps.io/Inference_mean/. Under the Confidence and Significance Tests tab:
- Select “Summary Statistics” from the drop-down menu under “Enter Data.”
- Type in “Calorie Content” for the name of the variable, and then enter the sample size and the sample mean and standard deviation for this example.
- For Type of Inference, select “Confidence Interval.”
- Use the slider for the confidence level to select a 95% confidence level.
c) Interpret the confidence interval from Part B.
d) An article on the website Global News included the following statement:
“Although every person’s daily caloric intake is individual, based on their personal goals and needs, nutrition experts estimate that average daily consumption at each meal should be broken down as follows: 300 to 400 calories for breakfast and 500 to 700 calories each for lunch and dinner. Snacks shouldn’t exceed 200 calories.”
Do you think that your confidence interval from Part B provides evidence that the mean calorie content of fast-food lunches does not meet the recommendations in the given quote? Explain.
- 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 ↵
