## Engage NY Eureka Math Algebra 2 Module 4 Lesson 15 Answer Key

### Eureka Math Algebra 2 Module 4 Lesson 15 Example Answer Key

Example 1:

A high school principal claims that 50% of the school’s students walk to school in the morning. A student attempts to verify the principal’s claim by taking a random sample of 40 students and asking them if they walk to school in the morning. Sixteen of the sampled students say they usually walk to school in the morning, giving a sample proportion of \(\frac{16}{40}\) = 0. 40, which seems to dispel the principal’s claim of 50%. But could the principal be correct that the proportion of all students who walk to school is 50%?

a. Make a conjecture about the answer.

Answer:

b. Develop a plan for how to respond.

Answer:

Help the student make a decision on the principal’s claim by investigating what kind of sample proportions you would expect to see if the principal’s claim of 50% is true. You will do this by using technology to simulate the flipping of a coin 40 times.

Exploratory Challenge 1/Exercises 1 – 9:

In Exercises 1 – 9, students should assume that the principal is correct that 50% of the population of students walk to school. Designate heads to represent a student who walks to school.

Exercise 1.

Simulate 40 flips of a fair coin. Record your observations in the space below.

Answer:

Answers will vary. Sample response:

T T T H H H T H H H T T T H T T H H H H T H H T H T H T T H T T T H T H T T T T

Exercise 2.

What is the sample proportion of heads in your sample of 40? Report this value to your teacher.

Answer:

Answers will vary. Sample response: \(\frac{18}{40}\) = 0.45

Exercise 3.

Repeat Exercises 1 and 2 to obtain a second sample of 40 coin flips.

Answer:

Answers will vary.

Your teacher will display a graph of all the students’ sample proportions of heads.

The following is an example of a sampling distribution of sample proportions of heads in 40 flips of a coin:

Exercise 4.

Describe the shape of the distribution.

Answer:

Answers will vary. The shape of the distribution shown above is slightly skewed.

Exercise 5.

What was the smallest sample proportion observed?

Answer:

Answers will vary. In the sample graph, 0.25

Exercise 6.

What was the largest sample proportion observed?

Answer:

Answers will vary. In the sample graph, 0.65

Exercise 7.

Estimate the center of the distribution of sample proportions.

Answer:

Answers will vary. In the sample graph, about 0.50

Your teacher will report the mean and standard deviation of the sampling distribution created by the class.

Answer:

Answers will vary. The mean will be approximately 0.5, and the standard deviation will be approximately 0.079. From the sample graph, the mean is 0.493, and the standard deviation is 0.085.

Exercise 8.

How does the mean of the sampling distribution compare with the population proportion of 0.50?

Answer:

Answers will vary. From the sample response, the population proportion of 0.50 is very close to the mean of the sampling distribution.

Exercise 9.

Recall that a student took a random sample of 40 students and found that the sample proportion of students who walk to school was 0.40. Would this have been a surprising result if the actual population proportion was 0.50 as the principal claims?

Answer:

Answers will vary. Based on the sample responses, the value of 0.40 is about one standard deviation from the mean. There were quite a few samples in the simulation that resulted in sample proportions that were 0.40 or smaller. Hence, a value of 0.40 would not be a surprising result if the population was 0.50.

Example 2: Sampling Variability

What do you think would happen to the sampling distribution you constructed in the previous exercises had everyone in class taken a random sample of size 80 instead of 40? Justify your answer. This will be investigated in the following exercises.

Answer:

Answers will vary. The results would be more accurate because there are more samples.

Exploratory Challenge 2/Exercises 10 – 22:

Exercise 10.

Use technology and simulate 80 coin flips. Calculate the proportion of heads. Record your results in the space below.

Answer:

Answers will vary. Sample response: \(\frac{39}{80}\) = 0.4875

Exercise 11.

Repeat flipping a coin 80 times until you have recorded a total of 40 sample proportions.

Answer:

Answers will vary. See Exercise .12 for o dot plot of the sampling distribution of the proportion of heads in 80 flips of a coin.

Exercise 12.

Construct a dot plot of the 40 sample proportions.

Answer:

Exercise 13.

Describe the shape of the distribution.

Answer:

Answers will vary. From the sample response, the distribution is symmetric and mound-shaped.

Exercise 14.

What was the smallest proportion of heads observed?

Answer:

Answers will vary. From the sample response, 0.39.

Exercise 15.

What was the largest proportion of heads observed?

Answer:

Answers will vary. From the sample response, 0.63.

Exercise 16.

Using technology, find the mean and standard deviation of the distribution of sample proportions.

Answer:

Answers will vary. The mean will be approximately 0. 5, and the standard deviation will be approximately 0.055. In the example above, the mean is 0.508, and the standard deviation is 0.061.

Exercise 17.

Compare your results with the others in your group. Did you have similar means and standard deviations?

Answer:

Answers will vary. All the groups should have similar means and standard deviations.

Exercise 18.

How does the mean of the sampling distribution based on 40 simulated flips of a coin (Exercise 1) compare to the mean of the sampling distribution based on 80 simulated coin flips?

Answer:

Both of the means will be approximately equal to 0.50.

Exercise 19.

Describe what happened to the sampling variability (standard deviation) of the distribution of sample proportions as the number of simulated coin flips increased from 40 to 80.

Answer:

The standard deviation decreased as the number of coin flips went from 40 to 80.

Exercise 20.

What do you think would happen to the variability (standard deviation) of the distribution of sample proportions if the sample size for each sample was 200 instead of 80? Explain.

Answer:

The standard deviation will decrease as the sample size increases.

Exercise 21.

Recall that a student took a random sample of 40 students and found that the sample proportion of students who walk to school was 0.40. If the student had taken a random sample of 80 students instead of 40, would this have been a surprising result if the actual population proportion was 0.50 as the principal claims?

Answer:

Answers will vary. The value of 0.40 is about two standard deviations from the mean. Only two of the 40 simulated samples resulted in a sample proportion of 0.40 or smaller. A sample proportion of 0.40 would be a fairly surprising result.

Exercise 22.

What do you think would happen to the sampling distribution you constructed in the previous exercises if everyone in the class took a random sample of size 80 instead of 40? Justify your answer.

Answer:

Answers will vary. The more samples, the more accurate the simulation will be because the standard deviation decreases as sample size increases.

### Eureka Math Algebra 2 Module 4 Lesson 15 Problem Set Answer Key

Question 1.

A student conducted a simulation of 30 coin flips. Below is a dot plot of the sampling distribution of the proportion of heads. This sampling distribution has a mean of 0.51 and a standard deviation of 0.09.

a. Describe the shape of the distribution.

Answer:

The distribution is approximately symmetric. Some students may respond that the distribution is slightly skewed to the left.

b. Describe what would have happened to the mean and the standard deviation of the sampling distribution of the sample proportions if the student had flipped a coin 50 times, calculated the proportion of heads, and then repeated this process a total of 30 times.

Answer:

The mean would be approximately equal to 0.51, and the standard deviation would be less than 0.09.

Question 2.

What effect does increasing the sample size have on the mean of the sampling distribution?

Answer:

Increasing the sample size does not affect the mean of the sampling distribution. The mean of the sampling distribution is approximately equal to the population mean for any sample size.

Question 3.

What effect does increasing the sample size have on the standard deviation of the sampling distribution?

Answer:

Increasing the sample size decreases the standard deviation of the sampling distribution.

Question 4.

A student wanted to decide whether or not a particular coin was fair (i.e., the probability of flipping a head is 0.5). She flipped the coin 20 times, calculated the proportion of heads, and repeated this process a total of 40 times. Below is the sampling distribution of sample propOrtions of heads. The mean and standard deviation of the sampling distribution are 0.379 and 0.091, respectively. Do you think this was a fair coin? Why or why not?

Answer:

If the coin was fair, the sampling distribution should be centered at about 0.50. Here, the sampling distribution is centered pretty far to the left of 0. 50. Hence, it is unlikely that the probability of heads for this coin would be 0.50.

Question 5.

The same student flipped the coin loo times, calculated the proportion of heads, and repeated this process a total of 40 times. Below is the sampling distribution of sample proportions of heads. The mean and standard deviation of the sampling distribution are 0.405 and 0.046, respectively. Do you think this was a fair coin? Why or why not?

Answer:

If the coin was fair, the sampling distribution should be centered at about 0.50. Here, the sampling distribution is centered pretty far to the left of 0.50. Hence, it is unlikely that the probability of heads for this coin would be 0.50.

### Eureka Math Algebra 2 Module 4 Lesson 15 Exit Ticket Answer Key

Question 1.

Below are three-dot plots of the proportion of tails in 20, 60, or 120 simulated flips of a coin. The mean and standard deviation of the sample proportions are also shown for each of the three-dot plots. Match each dot plot with the appropriate number of flips. Clearly explain how you matched the plots with the number of simulated flips.

a. Dot Plot 1

Mean: 0.502

Standard deviation: 0.046

Answer:

Sample Size: 120 flips of the coin

Explain: As the number of flips increases, the standard deviation decreases. The sampling distribution based on 120 flips has the smallest standard deviation.

b. Dot Plot 2

Mean: 0.518

Standard deviation: 0.064

Answer:

Sample Size: 60 flips of the coin

Explain: As sampling size increases, the standard deviation decreases. Because this sample size falls between the other two, its standard deviation will be between the standard deviations of the other sample sizes.

c. Dot Plot 3

Mean: 0498

Standard deviation: 0.110

Answer:

Sample Size: 20 flips of the coin

Explain: This standard deviation is the largest, which means that the sample size must be the smallest.