Calculate the mean and standard deviation of the sampling distribution of a sample proportion and interpret the standard deviation.
Determine if the sampling distribution of a sample proportion is approximately Normal.
If appropriate, use a Normal distribution to calculate probabilities involving a sample proportion .
Activity: What is the Proportion of Orange Reese's?
Students will be excited to eat some candy when they see the question for today’s Activity. Then you will crush their dreams by revealing the applet they will use to simulate taking samples of Reese’s Pieces.
We suggest doing page 1 of the activity as a whole class. The entire page is review from Chapter 6 and we want students to spend more time working and thinking on page 2 of the Activity.
You will need to prepare two posterboards for dotplots. The first will be the sampling distribution of X (number of successes) and the second will be the sampling distribution of phat (proportion of successes). Be sure to use the same scale on both…so the number of successes goes from 10 to 30 and the proportion of successes goes from 0.20 to 0.60. This will help to reveal to students that the shape of both is identical.
Population distribution, distribution of a sample, or a sampling distribution?
While the distinction between these three distributions is a Learning Target from Lesson 7.1, the Reese’s Pieces simulation provides a concrete visual representation of the differences. Don’t forget to ask “What does this dot represent?”
The link between binomial and sample proportions
This Activity makes the very clear connection between the binomial distribution from Chapter 6 and the sampling distribution of a sample proportion. Put simply, the binomial distribution shows the number of successes, while the sampling distribution shows the proportion of successes. So how do we turn the number of successes into the proportion of successes?
Answer: divide by n.
So take our formulas for mean and standard deviation from Chapter 6 and divide them by n and this will give us the formulas that we need for the sampling distribution of a sample proportion.
Preparing for inference
In Chapter 9, we will perform a one sample z test for a proportion. The standardized test statistic (which will lead us to the P-value) will be given by the following formula. Can you see how students are actually already doing this calculation?