The Sampling Distribution of a Sample Proportion (Lesson 7.3)
Chapter 7 - Day 3
Learning Targets
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Calculate the mean and standard deviation the sampling distribution of a sample proportion and interpret the standard deviation.
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Determine if the sampling distribution of p-hat is approximately normal.
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If appropriate, use a normal distribution to calculate probabilities involving p-hat.
Experience First
In this lesson, students will do some inferential thinking within the context of sample proportions. There is a very close connection with sample proportions and the binomial distributions of Chapter 6. Here is the connection:
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For binomial distributions, the random variable of interest is the count of the number of successes.
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For sample proportions, the random variable of interest is the proportion of successes.
Of course it is very easy to move from the count of number of successes to the proportion by simply dividing by the sample size (n). This is the critical piece of information that students will need to understand the new formulas in this lesson.
Be sure that you prepare a poster for the dotplot of sample proportions with 0.5 at the center (fair coin).
While students are working in groups, you will want to check in on their answers for #4 (mean and SD of the sample proportions) and support groups that need it.
Formalize Later
An important part of the debrief of the activity is to introduce the new notation of p-hat, which is a proportion calculated from a sample. Don’t forget to ask students “What does this dot represent?” when referring to the dotplot. The correct answer is “A sample of 30 coin tosses, and a sample proportion (p-hat) calculated from that sample.”
This lesson provides a rare opportunity to do some Algebra! When deriving the formula for the SD of a distribution of sample proportions, you can show students how to use Algebra to simplify the formula.
The Large Counts condition is not new for students. They learned about this in Lesson 6.5 for binomial distributions. It seems reasonable that if the distribution of counts is approximately normal, then the distribution of proportions will also be approximately normal. After all, we know that dividing a distribution by a constant does not change the shape of the distribution (Lesson 2.2). So the Large Counts condition applies to a distribution of sample proportions just as it did for a binomial distribution.
The last question in the activity provides an excellent opportunity to remind students about the idea of statistical significance (first introduced in Lesson 4.7) and the 5% cutoff rule for significance.
This is the part of the course where all the fundamental building blocks that students have been collecting are starting to all come together!