Lesson 6.5 – The Sampling Distribution of a Sample Mean
- Find the mean and standard deviation of the sampling distribution of a sample mean and interpret the standard deviation.
- Use a normal distribution to calculate probabilities involving the sample mean when sampling from a normal population.
Activity: How tall are we?
To complete this activity, you will need the data. Give groups the population data and point out that the dotplot at the top of the activity is the population distribution. Add a title to dotplot and ask students what each dot represents. This is an important distinction between population and sampling distributions. Groups will estimate the mean and standard deviation from the population distribution. You could make this a competition if you wanted. The mean is 168.96 cm and the standard deviation is 9.606 cm.
Groups need to collect samples of size 5 from the population and calculate the mean. When they’re done they should come draw a dot for each sample mean on the projected dotplot from the activity page. After all the sample means have been drawn, make sure a title has been added and again ask what each dot represents. They should realize that the dots no longer represent one student, but an average from a group of 5 students.
To compare the population distribution with the sampling distribution use Shape, Center and Spread as we did in chapter 1. This is where you need to add in the new formulas. For Shape, explain that a sampling distribution is approximately normal when the population distribution is approximately normal. By Center and Spread, add to the margin the new formulas for mean and standard deviation.
Since the formulas for mean and standard deviation for the sampling distribution of a sample mean are new, we want to really tie in the connection to the population distribution. Discuss how the new mean is approximately the same as the population mean but the standard deviation goes down. Both of those make sense since we learned in chapter 3 that when you increase sample size you decrease variability. When you’re done, add the new formulas to the formula review sheet.
If your students are confused by the pictures at the top of the activity, don’t hold it against them. They’re young. But please educate them on the greatness of The Princess Bride and Andre the Giant.