9C In-Class Activity

In 2017–2018, the National Health and Nutrition Examination Survey estimated that 42.4% of American adults fit the medical definition of obese.[1] A large medical clinic instituted a wellness  and nutrition program for its patients, where patients could opt-in to receive text messages with nutrition and exercise tips  or use an app to monitor their diets and activity levels. This clinic would like to  determine if, after a year of the program, the proportion of its patients who are obese is less than the national average.

Question 1

As the statistician on the project, explain how you could conduct a study to decide if  there is evidence that the rate of obesity among the clinic’s patients is less than 42.4%.

In this in-class activity, you will explore the sampling distribution of sample proportions  for varying values of the population proportion and sample size.

Go to the DCMP Sampling Distribution of the Sample Proportion tool at  https://dcmathpathways.shinyapps.io/SampDist_Prop/.

You will use this tool to simulate samples of different sizes from the American adult  population, where the sample proportion who are obese is calculated for each sample.

Question 2

You would like to simulate random samples of American adults and measure the proportion who are obese in each sample. What value should you set for the “Population Proportion ([latex]p[/latex])” in the data analysis tool?

Question 3

If you were to draw many random samples of American adults and measure the proportion of each sample who are obese, how would you predict the shape, center, and variability of the sampling distribution of sample proportions to change as the sample size increased?

Question 4

Set the population proportion ([latex]p[/latex]) in the tool to the value you specified in Question 2. You will need to check the “Enter Numerical Values for [latex]n[/latex] and [latex]p[/latex]” box to enter a value with more than two decimal places.

  1. Set the sample size to [latex]n= 1[/latex]. Then draw 1,000 random samples of size 1 from the population. Sketch a picture of the resulting sampling distribution of the sample proportion. Be sure to label your axes and provide a descriptive title in your sketch.
  2. Now set the sample size to [latex]n= 5[/latex] and click “Reset.” Draw 1,000 random samples of size 5 from the population. Sketch a picture of the resulting sampling distribution of the sample proportion. Be sure to label your axes and provide a descriptive title in your sketch.
  3. Now set the sample size to [latex]n= 25[/latex] and click “Reset.” Draw 1,000 random samples of size 25 from the population. Sketch a picture of the resulting sampling distribution of the sample proportion. Be sure to label your axes and provide a descriptive title in your sketch.
  4. Now set the sample size to [latex]n= 100[/latex] and click “Reset.” Draw 1,000 random samples of size 100 from the population. Sketch a picture of the resulting sampling distribution of the sample proportion. Be sure to label your axes and provide a descriptive title in your sketch.

Question 5

Consider the graphs you drew in Question 4.

  1. Explain what happened to the center of the sampling distribution as the sample size increased. Does this match your prediction from Question 3?
  2. Explain what happened to the variability of the sampling distribution as the sample size increased. Does this match your prediction from Question 3?
  3. Explain what happened to the shape of the sampling distribution as the sample size increased. Does this match your prediction from Question 3?

In the previous two questions, you witnessed the Central Limit Theorem at work. The Central Limit Theorem states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution. In In-Class Activity 9.B, we learned expressions for the mean and standard deviation of sample proportions. Combining all this together, we have the following result:

Sampling Distribution of the Sample Proportion

When taking many random samples of size [latex]n[/latex] from a population distribution with proportion [latex]p[/latex]:

The mean of the distribution of sample proportions is [latex]p[/latex].

The standard deviation of the distribution of sample proportions is [latex]\sqrt{\frac{p(1-p)}{n}}[/latex].

If [latex]np\geq 10[/latex] and [latex]n(1-p) \geq 10[/latex], then the Central Limit Theorem states that the distribution of the sample proportions follows an approximate normal distribution with mean p and standard deviation [latex]\sqrt{\frac{p(1-p)}{n}}[/latex]

Question 6

Based on the Central Limit Theorem, what is the approximate distribution of [latex]p[/latex], a sample proportion of American adults who are obese, when [latex]p= 0.424[/latex] and [latex]n= 100[/latex]?

Question 7

Go to the DCMP Normal Distribution tool at https://dcmathpathways.shinyapps.io/NormalDist/ and use the distribution described in Question 6 to approximate the following probabilities.

  1. What is the approximate probability that at most 35% of individuals in a random sample of 100 American adults are obese? Round to the nearest ten thousandth.
  2. What is the approximate probability that [latex]p> 0.45[/latex]?
  3. What is the probability that between 40% and 50% of individuals in a random sample of 100 American adults are obese?

Question 8

In the DCMP Sampling Distribution of the Sample Proportion tool (https://dcmathpathways.shinyapps.io/SampDist_prop/), set [latex]p= 0.424[/latex] and [latex]n= 100[/latex].

Then generate 10,000 random samples of size 100 from this population. Select the box next to “Find Probability for Samp. Dist.” under “Options” and use your simulated distribution to answer the following questions.

  1. What proportion of the simulated sample proportions is less than 0.35? How does this value compare to your answer in Question 7, Part a?Note: Enter 0.35 into the “At or below which value?” box and report the “Proportion of simulations resulting in sample proportion less than or equal to 0.35.”
  2. What proportion of the simulated sample proportions is greater than 0.45? How does this value compare to your answer in Question 7, Part b?
  3. What proportion of the simulated sample proportions is between 0.40 and 0.50? How does this value compare to your answer in Question 7, Part c?

Question 9

After a year of the wellness and nutrition program, the large medical clinic surveyed a random sample of 500 of its patients and found that 203 of the patients surveyed met the clinical definition of obese.

  1. Assume that the true proportion of the clinic’s patients who are obese is 0.424. What are the mean and standard deviation of the distribution of sample proportions when [latex]n= 500[/latex]?
  2. Assume that the true proportion of the clinic’s patients who are obese is 0.424. Explain why we can model the distribution of sample proportions from random samples of 500 patients with a normal distribution.
  3. Use the normal approximation for p to complete the following sentence:If the true proportion of the clinic’s patients who are obese is 0.424, then in approximately 95% of all random samples of 500 patients, the sample proportion who are obese would fall between _______ and _______.
  4.  Consider your answer to Part c. Do these data provide strong evidence that the percentage of patients at this clinic who are obese is less than the national average of 42.4%? Explain.

Question 10

The large medical clinic would like to do a follow-up study but would like the standard deviation of the sample proportion to be no higher than 0.01. Assume again that the true proportion of the clinic’s patients who are obese is 0.424. How many individuals would they need to ensure that the standard deviation of sample proportions is at most 0.01?

 

  1. Hales, C. M., Carroll, M. D., Fryar, C. D., & Ogden, C. L. (2020, February). . National Center for Health Statistics Data Brief No. 360. https://www.cdc.gov/nchs/products/databriefs/db360.htm ↵