Students calculate the mean of a sample collected in an earlier lesson to compare with their partners. Students experience firsthand that different samples from the same population can produce different results. In later activities students will use the data they have collected here to develop a deeper understanding of sampling variability.
Launch
Arrange students in groups of 2 so that different partners are used from the ones used in the earlier activity analyzing reaction times of 12th graders for a track meet.
Remind students that the numbers came from a survey of all 120 seniors from a certain school. The numbers represent their reaction time in seconds during an activity in which they clicked a button as soon as they noticed that a square changed color. Those 120 values are the population for this activity.
Give students 2 minutes of quiet work time followed by partner work time. Follow with a whole-class discussion.
Earlier, you worked with the reaction times of twelfth graders to see if they were fast enough to help out at the track meet. Look back at the sample you collected.
Calculate the mean reaction time for your sample.
Did you and your partner get the same sample mean? Explain why or why not.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of the discussion is for students to think about how the data they collected relates to the population data.
Some questions for discussion:
“Based on the information you currently know, estimate the mean of the population. Explain your reasoning.” (Since both my partner and I got means close to 0.4, I think the population mean will be a little greater than 0.4.)
“If each person selected 40 reaction times for their sample instead of 20, do you think this would provide a better estimate for the population mean?” (Since the sampling method is random, and thus fair, it should produce a better estimate.)
17.2
Activity
Standards Alignment
Building On
Addressing
7.SP.A
Use random sampling to draw inferences about a population.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
In this activity, a dot plot is created by the class that includes all of the calculated sample means. Students then compare that display to the data from the entire population to better understand the information that can be gained from a sample mean (MP7). In the discussion that follows the activity, students look at similar displays of sample means for samples of different sizes (MP8). Students should see that larger samples tend to more accurately estimate the population mean than smaller samples.
Launch
Display the basis for a dot plot of sample means for all to see.
Allow students 5 minutes to create the class dot plot and complete the first set of questions.
Ask students to include their sample means from the previous task to the display for all to see (since the samples originated in pairs, there will be many repeats. It is OK to include the repeats or ensure only 1 of the original partners includes their point on the display). They may do this by adding a dot to the display on the board, placing a sticky note in the correct place, or passing a large sheet of paper around the class.
After students have had a chance to answer the first set of questions, pause the class to display the dot plot of the the reaction times for the entire population of 120 seniors for all to see.
Provide students 5 minutes to complete the problems. Follow with a whole-class discussion.
Activity
None
Your teacher will display a blank dot plot.
Plot your sample mean from the previous activity on your teacher's dot plot.
What do you notice about the distribution of the sample means from the class?
Where is the center?
Is there a lot of variability?
Is it approximately symmetric?
The population mean is 0.445 seconds. How does this value compare to the sample means from the class?
Pause here so your teacher can display a dot plot of the reaction times of the population.
What do you notice about the distribution of the population?
Where is the center?
Is there a lot of variability?
Is it approximately symmetric?
Compare the two displayed dot plots.
Based on the distribution of sample means from the class, do you think the mean of a random sample of 20 items is likely to be:
within 0.01 seconds of the actual population mean?
within 0.1 seconds of the actual population mean?
Explain or show your reasoning.
Student Response
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Building on Student Thinking
The center of the dot plot created by the class can be thought of as a mean of sample means. The phrase can be difficult for students to think through, so remind them what the dot they added to the dot plot represents and how they calculated that value. Consider displaying the class dot plot for the rest of the unit to help students remember this example to understand similar dot plots.
Activity Synthesis
Display the dot plot of sample means for all to see throughout the unit. It may be helpful to refer to this display when viewing dot plots of sample means in later lessons.
The purpose of the discussion is for students to understand that the centers of the two dot plots in the activity are close, if not the same, but the sample means have less variability, and the shapes of the population distribution and the sample mean distribution are probably different. Additionally, students should use the dot plots shown here to see that larger samples have sample means that are still centered around the population mean, but have less variability than the means from smaller samples. Therefore, larger samples should provide better estimates of the population mean than smaller samples do.
Items for further discussion:
“If the sample size were increased to 30 instead of 20, what do you think the distribution might look like?” (There would be less variability in the same means, but they should be centered around the same value.)
For a different set of data representing the wingspan in centimeters of a certain species of bird, compare these dot plots.
Here is a dot plot of the population data.
Here is a dot plot of the sample means for 100 different random samples each of size 10.
Here is a dot plot of the sample means for 100 different random samples each of size 30.
“What do you notice about the distribution of the sample means as the sample size increases?” (The variability decreases, but the center stays in about the same place.)
“Use the dot plots to explain why the sample mean from a random sample of size 30 would be a better estimate of the population mean than the sample mean from a random sample of size 10.” (Since the variability is less, the sample mean is probably closer to the true mean with size 30 than with size 10.)
MLR8 Discussion Supports. Display sentence frames to support whole-class discussion: “When the sample size is 10, I noticed that the distribution . .. .” “When the sample size is 30, I noticed that the distribution . .. .” and “It’s better to have a larger/smaller sample size to estimate the population mean because . . . .” Advances: Conversing, Representing
Action and Expression: Internalize Executive Functions. To support organization, provide students with a graphic organizer to describe the connections between sample size and mean distribution shape. Supports accessibility for: Language, Organization
17.3
Activity
Standards Alignment
Building On
Addressing
7.SP.A.2
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
In this activity, students continue to look at how variability in the sample means can be used to think about the accuracy of an estimate of a population mean. If the means from samples tend to be very spread out, then there is reason to believe that the mean from a single sample may not be especially close to the mean for the population. If the means from samples tend to be tightly grouped, then there is reason to believe that the mean from a single sample is a good estimate of the mean for the population (MP7).
Launch
Arrange students in groups of 2.
Remind students about the context of earlier lessons in which they examined ages of people watching various shows.
Tell students that, although the dot plots seem to have a similar shape, attention should be given to the scale of the horizontal axis.
Allow students 3–5 minutes of quiet work time followed by partner discussion. Then follow with a whole-class discussion.
Representation: Develop Language and Symbols. Make connections between representations visible. Ask students to explain the meaning of a particular dot on one of the dot plots. Ensure students understand that a single dot represents a mean, not the age of one person. Supports accessibility for: Language, Conceptual Processing
Activity
None
Here are the mean ages for 100 different samples of viewers from each show.
A dot plot for “sample means for Trivia the Game Show” with the numbers 40 through 70, in increments of 2, is indicated. The data are as follows: 44 years old, 2 dots. 44 point 5 years old, 1 dot. 47 years old, 2 dots. 48 years old, 1 dot. 50 point 5 years old, 1 dot. 51 years old, 1 dot. 52 years old, 4 dots. 52 point 5 years old, 4 dots. 53 years old, 3 dots. 54 years old, 1 dot. 54 point 5 years old, 2 dots. 55 years old, 2 dots. 55 point 5 years old, 3 dots. 56 years old, 6 dots. 56 point 5 years old, 2 dots. 57 years old, 6 dots. 57 point 5 years old, 1 dot. 58 years old, 6 dots. 58 point 5 years old, 7 dots. 59 years old, 9 dots. 59 point 5 years old, 2 dots. 60 years old, 4 dots. 60 point 5 years old, 2 dots. 61 years old, 4 dots. 61 point 5 years old, 3 dots. 62 years old, 2 dots. 63 years old, 3 dots. 63 point 5 years old, 1 dot. 64 years old, 1 dot. 64 point 5 years old, 2 dots. 65 years old, 1 dot. 65 point 5 years old, 1 dot. 66 years old, 2 dots. 66 point 5 years old, 1 dot. 67 point 5 years old, 1 dot. 68 years old, 1 dot.
A dot plot for “sample means for Science Experiments YOU Can Do” with the numbers 9 through 14, in increments of zero point 5, indicated. The data are as follows: 9 point 8 years old, 1 dot. 10 point 2 years old, 1 dot. 10 point 3 years old, 1 dot. 10 point 4 years old, 1 dot. 10 point 6 years old, 1 dot. 10 point 7 years old, 2 dots. 10 point 8 years old, 1 dot. 10 point 9 years old, 3 dots. 11 years old, 1 dot. 11 point 1 years old, 1 dot. 11 point 2 years old, 2 dots. 11 point 3 years old, 1 dot. 11 point 4 years old, 3 dots. 11 point 5 years old, 1 dot. 11 point 6 years old, 5 dots. 11 point 7 years old, 5 dots. 11 point 8 years old, 4 dots. 11 point 9 years old, 8 dots. 12 years old, 6 dots. 12 point 1 years old, 7 dots. 12 point 2 years old, 3 dots. 12 point 3 years old, 4 dots. 12 point 4 years old, 9 dots. 12 point 5 years old, 2 dots. 12 point 6 years old, 8 dots. 12 point 7 years old, 3 dots. 12 point 8 years old, 3 dots. 12 point 9 years old, 5 dots. 13 years old, 1 dot. 13 point 1 years old, 1 dot. 13 point 3 years old, 1 dot. 13 point 4 years old, 1 dot. 13 point 5 years old, 1 dot. 13 point 6 years old, 1 dot. 13 point 8 years old, 2 dots.
A dot plot for “sample means for Learning to Read” with the numbers 5 through 7, in increments of zero point 2, indicated. The data are as follows: 5 years old, 1 dot. 5 point 1 years old, 1 dot. 5 point 2 years old, 1 dot. 5 point 3 years old, 4 dots. 5 point 4 years old, 4 dots. 5 point 5 years old, 6 dots. 5 point 6 years old, 8 dots. 5 point 7 years old, 5 dots. 5 point 8 years old, 10 dots. 5 point 9 years old, 10 dots. 6 years old, 10 dots. 6 point 1 years old, 10 dots. 6 point 2 years old, 9 dots. 6 point 3 years old, 4 dots. 6 point 4 years old, 8 dots. 6 point 5 years old, 4 dots. 6 point 6 years old, 2 dots. 6 point 7 years old, 2 dots. 6 point 8 years old, 1 dot.
For each show, use the dot plot to estimate the population mean.
Trivia the Game Show
Science Experiments YOU Can Do
Learning to Read
For each show, are most of the sample means within 1 year of your estimated population mean?
Suppose you take a new random sample of 10 viewers for each of the 3 shows. Which show do you expect to have the new sample mean closest to the population mean? Explain or show your reasoning.
Student Response
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Building on Student Thinking
Students may get confused about what the dot plots represent. Refer to the class dot plot of sample means from the reaction time activity to help them understand that a single dot on the dot plot represents a mean from a single sample. Each dot plot shows means from several different samples.
Activity Synthesis
The purpose of the discussion is to think about what the information in the given dot plots tells us about the accuracy of an estimate of a population mean based on the sample mean from a single sample.
Consider asking these questions for discussion:
“In the first dot plot for ‘Trivia the Game Show,’ what does the dot at 68 represent?” (There is a sample of 10 viewers that has a mean age of 68 years.)
“Based on the dot plot shown for ‘Learning to Read,’ do you think any 7 year olds are included in the data?” (Yes. Because one of the sample means is 6.8, there are probably a few 7 year olds and some 6 year olds in that sample.)
“Why is this kind of dot plot useful?” (It shows how variable the means from samples can be, so it gives a good idea of how accurate a sample mean might be as an estimate of a population mean.)
“Why might it be difficult to obtain a dot plot like this?” (Each dot represents a sample of 10. It might be hard to get 100 different samples of size 10.)
Lesson Synthesis
Consider asking these discussion questions to emphasize the main ideas of the lesson:
“In the first dot plot for Trivia the Game Show, what does the dot at 68 represent?” (There is a sample of 10 viewers that has a mean age of 68 years.)
“Based on the dot plot shown for Learning to Read, do you think any 7-year-olds are included in the data?” (Yes. Because one of the sample means is 6.8, there are probably a few 7-year-olds and 6-year-olds in that sample.)
“Why is this kind of dot plot useful?” (It shows how variable the means from samples can be, so it gives a good idea of how accurate a sample mean might be as an estimate of a population mean.)
“Why might it be difficult to obtain a dot plot like this?” (Each dot represents a sample of 10. It might be hard to get 100 different samples of size 10.)
Student Lesson Summary
A population of hummingbirds has a mean weight of 11.6 grams. This dot plot shows the weights of 18 hummingbirds selected from the population that have the same mean weight as the population. The mean weight is indicated with a triangle.
<p>A dot plot labeled “hummingbird weights in grams.” The numbers 8 through 16 are indicated. A triangle at approximately 11.6 is indicated. The data are as follows: 8 grams, 1 dot; 9 grams, 1 dot; 10 grams, 2 dots; 11 grams, 2 dots; 12 grams, 4 dots; 13 grams, 3 dots; 14 grams, 3 dots; 15 grams, 1 dot.</p>
20 more samples, each with 5 hummingbirds, are selected from the original population. For each sample, the mean of the 5 birds is calculated and plotted on this dot plot. The triangle represents the population mean of 11.6 grams. Notice that the sample means are fairly close to the population mean, but that they are not exactly the same.
<p>A dot plot for "means of samples of size 5" with the numbers 8 through 16 indicated. The data are as follows: 10, 1 dot. 10 point 3, 3 dots. 10 point 9, 1 dot. 11, 3 dots. 11 point 2, 1 dot. 11 point 4, 2 dots. 11 point 6, 3 dots. 11 point 8, 1 dot. 12, 2 dots. 12 point 2, 2 dots. 12 point 8, 1 dot. There is a triangle located at 11.6.</p>
What is different if more hummingbirds are included in the samples? This dot plot shows the means of 20 samples of 10 hummingbirds, selected at random. Notice that the means for these samples are generally better estimates for the population mean because they tend to be even closer to the mean for the entire population.
<p>A dot plot labeled “means of samples of size 10.” The numbers 8 through 16 are indicated. The data are as follows: 10 point 9, 1 dot. 11, 1 dot. 11 point 1, 1 dot. 11 point 2, 1 dot. 11 point 3, 2 dots. 11 point 4, 2 dots. 11 point 5, 1 dot. 11 point 6, 3 dots. 11 point 7, 2 dots. 11 point 8, 1 dot. 11 point 9, 1 dot. 12 point 1, 1 dot. 12 point 3, 2 dots. 12 point 5, 1 dot. A triangle is located at 11 point 6.</p>
In general, as the sample size gets bigger, the mean of a sample is more likely to be closer to the mean of the population.
Standards Alignment
Building On
6.SP.A.3
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
