Engage NY Eureka Math Algebra 2 Module 4 Lesson 19 Answer Key
Eureka Math Algebra 2 Module 4 Lesson 19 Exercise Answer Key
Exercises 1 – 6: SAT Scores
SAT scores vary a lot. The table on the next page displays the 506 scores for students in one New York school district for a given year.
Table 1: SAT Scores for District Students
a. Looking at the table, how would you describe the population of SAT scores?
It is hard to tell. I see some numbers as low as 198 and others in the 700’s. You cannot tell much from just looking at the individual numbers.
b. Jason used technology to draw a random sample of size 20 from all of the scores and found a sample mean of 487. What does this value represent in terms of the graph below?
Random Sample from District SAT Scores
This represents the average SAT score for the sample and indicates where the scores in the dot plot are centered. If you computed the mean of the values for the SAT scores in the sample (i.e., one student had an SAT score of about 350, two had scored a bit over 350, one student scored about 625, and so on), you would find the mean SAT scores for those students to be 487.
If you were to take many different random samples of 20 from this population, describe what you think the sampling distribution of these sample means would look like.
The sampling distribution of these samples may be centered on a value close to 487. The spread may be similar to the example above, possibly from 350 to 600.
Everyone in Jason’s class drew several random samples of size 20 and found the mean SAT score. The plot below displays the distribution of the mean SAT scores for their samples.
a. How does the simulated sampling distribution compare to your conjecture in Exercise 2? Explain any differences.
I estimated the mean SAT score to be a bit higher than it seems to be from the simulated distribution, but my estimate was not that far off. I thought the spread would range from 350 to 600 like in Jason’s sample, but that was not a good estimate. I was thinking of the individual SAT scores in the sample and mixing them up with the means of samples of 20 scores, which is why my estimate was off.
b. Use technology to generate many more samples of size 20, and plot the means of those samples. Describe the shape of the simulated distribution of sample mean SAT scores.
Answers will vary. The following displays an example of a simulated distribution of sample means from random samples from district SAT scores:
The distribution is normal The mean of the sample mean SAT scores appears to be around 480, and the sample mean scores range from a little over 420 to about 530.
c. How did the simulated distribution using more samples compare to the one you generated in Exercise 3?
The maxima and minima were nearly the same in both of the distributions, 420 to 530 and a little over 420 to 520. The mean SAT scores of the simulated distributions of sample means in both seemed to be about 480.
d. What are the mean and standard deviation of the simulated distribution of the sample mean SAT scores you found in part (b)? (Use technology and your simulated distribution of the sample means to find the values.)
The mean SAT score of the simulated distribution of sample means is 478. The standard deviation is 21.5.
e. Write a sentence describing the distribution of sample means that uses the mean and standard deviation you calculated in part (d).
Almost all of the SAT scores are within two standard deviations from the mean, from 435 to 521.
Reflect on some of the simulated sampling distributions you have considered in previous lessons.
a. Make a conjecture about how you think the size of the sample might affect the distribution of the sample SAT means.
The larger the sample size, the smaller the spread of the distribution of sample means.
b. To test the conjecture, investigate the sample sizes 5, 10, 40, 50, and the simulated distribution of sample means from Exercise 3. Divide the sample sizes among your group members, and use technology to simulate sampling distributions of mean SAT scores for samples of the different sizes. Find the mean and standard deviation of each simulated sampling distribution.
The following represent simulated distributions of sample means of district SATscores for different size random samples:
c. How does the sample size seem to affect the simulated distributions of the sample SAT mean scores? Include the simulated distribution from part (b) of Exercise 3 in your response. Why do you think this is true?
As the sample size increases, the spread decreases. The standard deviation went from 40 for a sample of size 5 to about 14 for a sample of size 50. The means of the sampling distributions of mean SAT scores varied from 468 for the distribution for samples of size 10 to 478 for samples of size 20.
I would expect that a bigger sample would be more likely to look a lot like the population, and so bigger samples would not tend to be as different from one another as smaller samples. Because of this, the sample means would not differ as much from sample to sample for bigger samples.
a. For each of the sample sizes, consider how the standard deviation seems to be related to the range of the sample means in the simulated distributions of the sample SAT means you found in Exercise 4.
In each case, nearly all of the sample means are within two standard deviations of the mean or are a normal distribution.
b. How do your answers to part (a) compare to the answers from other groups?
Everyone had simulated sampling distributions that looked fairly alike and were centered in about the same place. The sample means for each sampling distribution were typically within two standard deviations of the mean of the simulated sampling distributions. The distributions were normal.
a. Make a graph of the distribution of the population consisting of the SAT scores for all of the students.
Possible response below of distribution of the SAT scores for district students:
b. Find the mean of distribution of SAT scores. How does it compare to the mean of the sampling distributions you have been simulating?
The mean SAT score for the students in the district is 475. 1 or 475. This is close to the means of the sampling distributions, even for fairly small samples.
Eureka Math Algebra 2 Module 4 Lesson 19 Problem Set Answer Key
Which of the following will have the smallest standard deviation? Explain your reasoning.
Sampling distribution of sample means for samples of size:
The largest sample size, 100, will have the smallest standard deviation because as the sample size increases, the variability in the sample mean decreases.
In light of the distributions of sample means you have investigated in the lesson, comment on the statements below for random samples of size 20 chosen from the district SAT scores.
a. Josh claimed he took a random sample of size 20 and had a sample mean score of 320.
A mean score of 320 seems very unlikely. None of the samples we have investigated had a sample mean score that low.
b. Sarfina stated she took a random sample of size 20 and had a sample mean of 520.
This seems plausible for the simulated distributions of sample mean scores; 520 was high, but still some of the random samples had mean scores greater than 520.
c. Ana announced that it would be pretty rare for the mean SAT score in a random sample to be more than three standard deviations from the mean SAT score of 475.
Given that the sample means in nearly all of the simulated distributions of the sample means were usually within two standard deviations from the mean, Ana is correct. It could happen, but it would not be usual.
Refer to your answers for Exercise 4, and then comment on each of the following:
a. A random sample of size 50 produced a mean SAT score of 400.
A mean score of 400 was less than any of the sample means in the simulated sampling distribution of sample means for samples of size 50, so this seems unlikely.
b. A random sample of size 10 produced a mean SAT score of 400.
A mean score of 400 was within two standard deviations of the mean for random samples of size 10, so it could have come from one of the samples.
c. For what sample sizes was a sample mean SAT score of 420 plausible? Explain your thinking.
A sample mean of 420 occurred in the simulated sampling distributions for samples of sizes 5 and 10 but not at all in the simulated distributions for samples of sizes 20, 40, and 50. So, it seems like 420 was a plausible outcome for samples of sizes 5 and 10.
Explain the difference between the sample mean and the mean of the sampling distribution.
Each sample of SAT scores had a mean SATscore, which is the sample mean. Then, all of those sample means formed a distribution of sample means, and we found the mean of that set, the mean of the sampling distribution of the sample mean – the mean of the means of the different samples.
Eureka Math Algebra 2 Module 4 Lesson 19 Exit Ticket Answer Key
Describe the difference between a population distribution, a sample distribution, and simulated sampling distribution, and make clear how they are different.
The distribution of the elements in a population (the SAT scores for students in a district) is a population distribution; the distribution of the elements in a random sample from that population (a subset of a given size chosen at random from the SAT scores) is a sample distribution; a simulated distribution of sample means for many random samples of a given size chosen from the population (the means of different random samples of the same size of students’ SAT scores) is a simulated sampling distribution.
Some students might also suggest that the meaning of sampling distribution of all samples is the samples of a given size selected from a population. This would be the distribution of the means of every possible sample that might be chosen.
Use the standard deviation and mean of the sampling distribution to describe an interval that includes most of the sample means.
Sample response: Typically, most of the means of the different random samples of the same size chosen from a population will be within two standard deviations of the mean or the mean ±2 standard deviations.