12B Preview

Preparing for the next class

In the next in-class activity, you will need to calculate and interpret a standardized sample mean, calculate and interpret the standard deviation of a sample mean, and calculate probabilities involving standardized statistics.

Go to the DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool at https://dcmathpathways.shinyapps.io/SampDist_cont/. You will use this data analysis tool to simulate random samples of colleges and examine the mean annual cost of attendance for each sample. Enter the following inputs:

Select Population Distribution: Real Population Data
Select Example: College Cost^[1]

The population distribution shown is the distribution of the average annual costs of attending college (in U.S. dollars) for a population of 1,909 public and private four-year U.S. colleges.

Question 1

1) What shape is the population distribution of the average annual costs of attending college?

Question 2

2) Generate 1,000 random samples of size [latex]n[/latex] = 20.

a) The histogram labeled “Data Distribution (Histogram from last generated sample)” is a histogram of the last randomly generated sample. What is the sample mean for this sample?

b) The same sample mean generated from the last randomly generated sample should be shown on the sampling distribution of the sample mean.

Use the mean and standard deviation of the simulated sampling distribution of the sample mean (from the 1,000 simulated samples) to calculate a z score for the last observed sample mean. Write a sentence interpreting this value.

c) Write a sentence interpreting the value of the standard deviation of the sampling distribution of the sample mean that you used to calculate the z score in Part B.

When we calculate a z-score for a statistic, as in Question 1, Part B, we call this a standardized statistic. Question 1 used simulation to estimate the mean and standard deviation of the sample mean. Mathematically, though, we know the exact formulas for these values. The standardized sample mean is

[latex]z=\frac{\bar{x}-[mean\;of\;\bar{x}]}{[std.\;deviation\;of\;\bar{x}]}=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}[/latex]

where [latex]\bar{x}[/latex] is the sample mean, [latex]\mu[/latex] is the population mean, [latex]\sigma[/latex] is the population standard deviation, and [latex]n[/latex] is the sample size. The statistic is “standardized” since it is centered to have a mean of 0 and scaled to have a standard deviation of 1.

If the population distribution is normal or the sample size is sufficiently large, this standardized statistic will follow a standard normal distribution—a normal distribution with a mean of 0 and a standard deviation of 1.

Question 3

3) What are the population mean and population standard deviation of the variable average annual cost of attending college? Use appropriate statistical notation.

Hint: These values are displayed in the main title of the Population Distribution in the tool.

Question 4

4) Using the values from Question 3, calculate the mean and standard deviation of the sampling distribution of the sample mean for samples of size [latex]n[/latex] = 20.

Question 5

5) Using your answers from Question 4, calculate the value of the standardized statistic for your observed sample mean from Question 2, Part A. How does this value compare to your answer from Question 2, Part B?

Question 6

6) Now, go to the DCMP Normal Distribution tool at https://dcmathpathways.shinyapps.io/NormalDist/. Select the Find Probability tab.

a) Enter your mean and standard deviation from Question 4 for the mean and standard deviation of the normal distribution. Calculate the probability of observing the value of your observed sample mean from Question 2, Part A or something smaller.

b) Enter the mean and standard deviation of a standard normal distribution. What is the probability of observing the value of your standardized statistic from Question 5 or something smaller? How does this probability compare to the probability from Part A?