Question

A Sampling Distribution for Statistics Graduate Programs Exercise 3.31 introduced the dataset StatisticsPhD, which gives enrollment for all 82 graduate statistics programs in the US in \(2009 .\) Use StatKey or other technology to generate a sampling distribution of sample means using a sample size of \(n=10\) from the values in this dataset. What shape does the distribution have? Approximately where is it centered? What is the standard error (in other words, what is the standard deviation of the sample means)?

Short answer

The distribution shape, center, and standard error will depend on the specific dataset values. The shape would likely be approximately normal (due to large repetition), the center is the mean of the sample means, and the standard error is the standard deviation of the sample means.

Step-by-step solution

Generate the Sampling Distribution

The first step is to generate a sampling distribution of sample means, using a sample size of \(n = 10\). You need to randomly select 10 data points from the dataset and calculate their mean. Repeat this process a large number of times (at least 1000 times) to obtain the sampling distribution.

Identify the Shape of the Distribution

After generating the sampling distribution, the next step is to identify its shape. You need to plot a histogram or a similar visual representation. Observe the shape of the distribution. It can be normal, skewed, bimodal, etc. Given the Central Limit Theorem, it's expected to have an approximate normal distribution for large enough repetitions.

Determine the Center of the Distribution

The center of a distribution is typically given by its mean or median. In this case, you need to calculate the average of the sample means obtained in step 1.

Calculate the Standard Error

The standard error is the standard deviation of the sample means. After having computed all the sample means in Step 1, calculate the standard deviation among them to get the standard error.