 # Understanding Variability when Estimating a Population Proportion

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Examples, videos, and solutions to help Grade 7 students learn how to understand the term sampling variability in the context of estimating a population proportion.

### New York State Common Core Math Grade 7, Module 5, Lesson 19

Download worksheets for Grade 7, Module 5, Lesson 19

### Lesson 19 Student Outcomes

• Students understand the term sampling variability in the context of estimating a population proportion.
• Students know that increasing the sample size decreases sampling variability.

### Lesson 19 Summary

• The sampling distribution of the sample proportion is a graph of the sample proportions for many different samples.
• The mean of the sample proportions will be approximately equal to the value of the population proportion.
• As the sample size increases the sampling variability decreases.

Lesson 19 Classwork

Question 1: How many hours of sleep do seventh graders get on a night in which there is school the next day?
Question 2: What proportion of the students in our school participate in band or orchestra?
Question 3: What is the typical weight of a backpack for students at our school?
Question 4: What is the likelihood that voters in a certain city will vote for building a new high school?
In a previous lesson, you selected several random samples from a population. You recorded values of a numerical variable. You then calculated the mean for each sample, saw that there was variability in the sample means, and created a distribution of sample means to better see the sampling variability. You then considered larger samples and saw that the variability in the distribution decreased when the sample size increases. In this lesson, you will use a similar process to investigate variability in sample proportions.

Example 1: Sample Proportion
Your teacher will give your group a bag that contains colored cubes, some of which are red. With your classmates, you are going to build a distribution of sample proportions.

Exercises 1–6
1. Each person in your group should randomly select a sample of 10 cubes from the bag. Record the data for your sample in the table below.
2. What is the proportion of red cubes in your sample of 10?
This value is called the sample proportion. The sample proportion is found by dividing the number of “successes” (in this example, the number of red cubes) by the total number of observations in the sample.
3. Write your sample proportion on a post-it note and place it on the number line that your teacher has drawn on the board. Place your note above the value on the number line that corresponds to your sample proportion.
The graph of all the students' sample proportions is called a sampling distribution of the sample proportions.
4. Describe the shape of the distribution.
5. Describe the variability in the sample proportions.
6. Based on the distribution, answer the following:
a. What do you think is the population proportion?
b. How confident are you of your estimate?

Example 2: Sampling Variability
What do you think would happen to the sampling distribution if everyone in class took a random sample of 30 cubes from the bag? To help answer this question you will repeat the random sampling you did in Exercise 1, except now you will draw a random sample of 30 cubes instead of 10.

Exercises 7–15
What do you think would happen to the sampling distribution if everyone in class took a random sample of 30 cubes from the bag? To help answer this question you will repeat the random sampling you did in Exercise 1, except now you will draw a random sample of 30 cubes instead of 10.
7. Take a random sample of 30 cubes from the bag. Carefully record the outcome of each draw.
8. What is the proportion of red cubes in your sample of 30?
9. Write your sample proportion on a post-it note and place the note on the number line that your teacher has drawn on the board. Place your note above the value on the number line that corresponds to your sample proportion.
10. Describe the shape of the distribution.
11. Describe the variability in the sample proportions.
12. Based on the distribution, answer the following:
a. What do you think is the population proportion?
b. How confident are you of your estimate?
c. If you were taking a random sample of 30 cubes and determined the proportion that was red, do you think your sample proportion will be within 0.05 of the population proportion? Explain.
13. Compare the sampling distribution based on samples of size 10 to the sampling distribution based on samples of size 30.
14. As the sample size increased from 10 to 30 describe what happened to the sampling variability of the sample proportions.
15. What do you think would happen to the variability of the sample proportions if the sample size for each sample was 50 instead of 30? Explain.

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