Estimating a Proportion (Lesson 8.3)
Chapter 8 - Day 4
Check the Random and Large Counts conditions for constructing a confidence interval for a population proportion.
Determine the critical value for calculating a C% confidence interval for a population proportion using Table A or technology.
Calculate a C% confidence interval for a population proportion.
This is always one of the best Activities of the year! If you are feeling brave, we suggest a "fake fall" in front of the whole class, where Hershey Kisses are scattered all over the floor. As you are slowly getting up from the terrible fall, you say "Wow...do you notice how some of the Kisses landed on their side and some landed flat. I wonder what proportion of the Kisses should land flat?"
This Activity is also exciting because students are able to put together all the puzzle pieces they have been learning up to this point:
In Chapter 2, students learned about the 68-95-99.7 rule for Normal Distributions. They learned that 95% of the data in a normal distribution is within 2 standard deviations of the mean (yes, we lied to them…we know better now to use 1.96). This idea becomes the critical value for a confidence interval.
In Chapter 4, students learned that the purpose of a random sample is so that we can generalize to the larger population. In this chapter, this idea becomes the Random condition for inference.
In Chapter 6, students learned to check the Large Counts condition in the binomial setting to be sure that the binomial distribution could be modeled with a Normal distribution.
In Chapter 7, students realized the same Large Counts condition can be used for the sampling distribution of a sample proportion. They also discovered a formula for the standard deviation of a sampling distribution of a sample proportion, which will become the standard error in this lesson (more below).
In question #5 in the activity, students are asked to calculate the standard deviation of the sampling distribution of a sample proportion. This is actually impossible because they would need to know p, the true proportion of Hershey Kisses that land flat, which is unknown. But they do have a good guess - the sample proportion from their sample of 50. So now students are actually just estimating the standard deviation, which is why we now call this the standard error.
In the debrief for question #6 in the activity, show students how to use Table A to find critical values. This also means that you will have to reveal that you had been lying to them all year with the 95% from the empirical rule. We suggest working through z* for 90%, 95%, and 99%, the most commonly used confidence levels.
Notice in the QuickNotes that we are giving 2 versions of the formula for a confidence interval for a proportion. We want students to get in the habit of writing these two formulas before they plug numbers in to get an answer. We call them the "General formula" and the "Specific formula". The nice thing about the general formula is that is is the same for any confidence interval we will calculate in this course.
Consider using the Confidence Intervals for Proportions applet to show students what would happen if we did many, many samples of Hershey Kisses and calculated a 95% confidence interval for each sample. Use the results to remind students of the interpretation of the confidence level from Lesson 8.2.