Using Sample Data to Decide if Two Population Means Are Different

Videos, examples, and solutions to help Grade 7 students learn how to express the difference in sample means as a multiple of a measure of variability.

New York State Common Core Math Grade 7, Module 5, Lesson 22

Worksheets for Grade 7

Lesson 22 Student Outcomes

• Students express the difference in sample means as a multiple of a measure of variability.
• Students understand that a difference in sample means provides evidence that the population means are different if the difference is larger than what would be expected as a result of sampling variability alone.br>

Lesson 22 Summary

Variability is a natural occurrence in data distributions. Two data distributions can be compared by describing how far apart their sample means are. The amount of separation can be measured in terms of how many MADs separate the means. (Note that if the two sample MADs differ, the larger of the two is used to make this calculation.)

Lesson 22 Classwork

In previous lessons, you worked with one population. Many statistical questions involve comparing two populations. For example: • On average, do boys and girls differ on quantitative reasoning?
• Do students learn basic arithmetic skills better with or without calculators?
• Which of two medications is more effective in treating migraine headaches?
• Does one type of car get better mileage per gallon of gasoline than another type?
• Does one type of fabric decay in landfills faster than another type?
• Do people with diabetes heal more slowly than people who do not have diabetes?
In this lesson, you will begin to explore how big of a difference there needs to be in sample means in order for the difference to be considered meaningful. The next lesson will extend that understanding to making informal inferences about population differences.

Example 1
Tamika’s mathematics project is to see whether boys or girls are faster in solving a KenKen-type puzzle. She creates a puzzle and records the following times that it took to solve the puzzle (in seconds) for a random sample of 10 boys from her school and a random sample of 11 girls from her school:

Exercises 1–3

1. On the same scale, draw dot plots for the boys’ data and for the girls’ data. Comment on the amount of overlap between the two dot plots. How are the dot plots the same, and how are they different?
   
2. Compare the variability in the two data sets using the MAD (mean absolute deviation). Is the variability in each sample about the same? Interpret the MAD in the context of the problem.
   
3. In the previous lesson, you learned that a difference between two sample means is considered to be meaningful if the difference is more than what you would expect to see just based on sampling variability. The difference in the sample means of the boys’ times and the girls’ times is 4.1 seconds (39.4 seconds - 35.3 seconds). This difference is approximately MAD.
   a. If 4 sec. is used to approximate the values of MAD (mean absolute deviation) for both boys and for girls, what is the interval of times that are within MAD of the sample mean for boys?
   b. Of the 10 sample means for boys, how many of them are within that interval?
   c. Of the 11 sample means for girls, how many of them are within the interval you calculated in part (a)?
   d. Based on the dot plots, do you think that the difference between the two sample means is a meaningful difference? That is, are you convinced that the mean time for all girls at the school (not just this sample of girls) is different from the mean time for all boys at the school? Explain your choice based on the dot plots.
   

Example 2
How good are you at estimating a minute? Work in pairs. Flip a coin to determine which person in the pair will go first. One of you puts your head down and raises your hand. When your partner says “start,” keep your head down and your hand raised. When you think a minute is up, put your hand down. Your partner will record how much time has passed. Note that the room needs to be quiet. Switch roles except this time you talk with your partner during the period when the person with his or her head down is indicating when they think a minute is up. Note that the room will not be quiet.

Exercises 4–7
Use your class data to complete the following.
4. Calculate the mean minute time for each group. Then find the difference between the “quiet” mean and the “talking” mean.
5. On the same scale, draw dot plots of the two data distributions and discuss the similarities and differences in the two distributions.
6. Calculate the mean absolute deviation (MAD) for each data set. Based on the MADs, compare the variability in each sample. Is the variability about the same? Interpret the MADs in the context of the problem.
7. Based on your calculations, is the difference in mean time estimates meaningful? Part of your reasoning should involve the number of MADs that separate the two sample means. Note that if the MADs differ, use the larger one in determining how many MADs separate the two means.

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