The goal of this discussion is for students to think about a distribution of sample means. Collect all of the means from the class, and create a dot plot. If it does not come up, ask students to describe the distribution shape. It should be approximately normal (bell-shaped and symmetric around 3.5).

Here are some questions for discussion:

• “Does the distribution support a claim that the mean is approximately 3.5?” (Yes, the center of the distribution is near 3.5.)
• “Which interval contains most of the means for your group’s data?” (Most of our means are between 2.8 and 4.2.)
• “What percentage of the means are in that interval?” (21 out of 28, or approximately 75%)
• “Estimate the percentage of the class data that is in this interval.”
• “How does the class distribution compare to your group’s distribution?”

If students get confused about the mean of the mean values, consider saying:

• “Tell me more about what the given numbers represent in the context of the problem.”

• “Describe a simulation that might result in those numbers.”

The goal of this discussion is for students to see that a population mean can be estimated from a sample and the standard deviation from a simulated sample distribution can provide a margin of error for the estimated population mean. The process is very similar to the one used for proportions earlier in the section.

Ask previously identified students, “How is finding the margin of error for a population mean estimate similar to finding the margin of error for a population proportion estimate?” (In both situations, we use data from a sample to run a simulation and collect the statistic from each simulated sample to create a sampling distribution. The margin of error for the point estimate is twice the standard deviation of the sampling distribution.)

Here are some questions for discussion:

• “What does the margin of error tell us?” (The margin of error tells us that the population mean has a 95% chance of being in the interval from the mean minus the margin of error to the mean plus the margin of error.)
• “What interval is likely to contain the mean number of the diameters of baseballs from that company?” (between 6.86 cm and 7.74 cm)
• “Can you think of another situation where you would want to estimate a population mean from a sample?” (I think it would be good for estimating the weight of products, like boxes of cereal. A box of cereal probably does not weigh exactly what the box says, but rather within a certain margin of error of that amount.)
• “Do you think that increasing the size of a sample would reduce the margin of error? Explain your reasoning.” (I think it would reduce the margin of error because a larger sample should display less variation from sample to sample.)