In the next preview assignment and in the next class, you will need to generate random  samples from a population, find values of the sample mean and sample standard deviation, and calculate z-scores and another similar standardized score for sample  means.

Calculating and Interpreting z-scores of Sample Means

Data collected by the Centers for Disease Control and Prevention show that the  average birthweight for babies in the United States is 7.17 pounds and the standard  deviation of birthweights is 1.30 pounds.^[1] Assume that birthweights in the United States follow an approximate normal distribution.

Go to the DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool at https://dcmathpathways.shinyapps.io/SampDist_cont/. You will use this tool to  simulate random samples of births and examine the mean birthweight for each sample.

Enter the following inputs:

• Select Population Distribution: Bell-Shaped
• Enter 7.17 and 1.30 for the population mean and standard deviation, respectively. (You will need to select the “Enter values for [latex]\mu[/latex] and [latex]\sigma[/latex]” option.)

The population distribution shown is our model for the distribution of all the birthweights  in the United States.

Because the birthweights of different babies will vary from sample to sample, the  sample mean birthweights will also vary from sample to sample. The tendency of  samples to have different statistics (means, proportions) than the population as a whole  due to randomness is called sampling variability, and the distribution of these  statistics is called a sampling distribution.

In the case of sample means, if we sample from a normal population as the one seen  here, the sampling distribution of the sample means will also have a normal distribution.  If the mean and standard deviation of the population are µ and σ, respectively, the mean  and standard deviation of the sample means for random samples of size [latex]n[/latex] are:

Mean of the sample means [latex]=\mu[/latex]

Standard deviation of the sample means = [latex]\frac{\sigma}{\sqrt{n}}[/latex]

Recall that a z-score of a value is calculated by subtracting the mean and then dividing by the standard deviation. Similarly, we can calculate the z-score of a sample mean by:

*missing latex* (not missing, but not working)

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