Grade 7

End-of-Unit Assessment

Elena would like to know the average speed that people drive on her street. With a speedometer, she tracks the speed of 30 random cars that drive through. The mean of Elena’s sample is 30 miles per hour, and the MAD (mean absolute deviation) is 3 miles per hour.

Select all the true statements.

Elena would be more likely to get an accurate estimate of the mean speed of the population by sampling 60 cars instead of sampling 30 cars.

The median speed of the sample must be between 25 and 35 miles per hour.

Another random sample of 30 cars is likely to have a mean between 24 and 36 miles per hour.

The mean speed of these 30 cars is likely to be exactly the same as the mean speed of a second sample of 30 cars.

The mean speed of these random 30 cars is likely to be exactly the same as the mean of all cars that drive in a residential area.

Answer: A, C

Teaching Notes

Students who select choice B are incorrectly using the MAD and mean information applied to the median. Students who do not select choice C have not learned this unit’s concept about how to use MAD to determine expected variation. Students who select choice E are making an error about the sample as it relates to the population, while students who select choice D are making a similar error about the nature of multiple samples. Students who do not select choice A may need a refresher that the mean of a sample is more likely to be close to the mean of the population as the sample size gets bigger.

Here is a dot plot showing how many books are read by each student per month.

<p>A dot plot. Number of books read by each student.</p>

Which of these is a representative sample of the population?

Answer:

Teaching Notes

Students who select choice A may believe that samples must have a symmetric distribution, which is not true. Students who select choice B or choice D do not understand how to show data in a sample population given the original.

The backpack weights for a random sample of 55 students in grade 6 in the United States is shown in the dot plot.

<p>Dot plot from 0 to 16, by 2’s. backpack weight in kilograms. </p><br>

Select all of the information that would be reasonable to estimate from this sample.

The median backpack weight for all grade 6 students in the United States is about 3 kg.

The median backpack weight for all grade 8 students in the United States is about 3 kg.

The mean backpack weight for all grade 6 students in the United States is about 3.6 kg.

The mean backpack weight for all grade 6 students at Sunnyside Middle School is 3.6 kg.

There are no grade 6 students with backpacks that weigh 12 kgs in the United States.

Answer: A, C

Teaching Notes

Students who select choice B are not recognizing that generalizations for information from this sample must apply to populations that include this sample. Students who select choice E may not recognize this as a sample or that there are likely more extreme values in the population as a whole.

A school plans to start selling snacks at their basketball games. They want to know which snacks would be most popular.

What is the population for the school's question?
Give an example of a sample the school could use to help answer their question that would be representative of the population.

Answer:

Sample response:

The population is basketball players, coaches, parents, and other spectators.
Select 30 people at the next basketball game at random.

Teaching Notes

This item checks whether students understand the meanings of the terms “population” and “sample” as well as their uses in context. For the first question, the answer must acknowledge all the people that attend school basketball games, not just students. There is lots of flexibility in acceptable responses to the second question since the item doesn't say how many people usually attend the games, but it should mention a random process.

The principals at 2 schools want to compare the number of minutes students spend on homework each evening. Each principal collects data from one class at their school to prepare information to share. Here is what they brought.

School A:

School B:

Mean: 47.3
Median: 45
MAD: 15.2
IQR: 25

In which school do students spend less time on their homework?
Which school has more variability in the amount of time spent on homework?

Answer:

School B
School A

Teaching Notes

Some students choose to make a box plot for School B. Although this is a short answer problem, watch for students mistakenly using the mean instead of the median or not using measures of variability correctly.

A politician is interested in how well he will do in the next election. He walks around and asks 10 people if they are voting for him in the next election. Here are the results:

Based on the data from these 10 people, estimate the proportion of all voters who will vote for this politician.
When the results from the election come out, this politician has lost with only 20% of the vote. What might explain the difference in the results? How could this politician collect a better sample if he runs again?

Answer:

Sample response:

$\frac{7}{10}$ or 0.7. (Any answer between 0.65 and 0.75 is reasonable.)
The sample was not selected using a random process. The politician should try to select people for the survey using more randomness to reduce bias.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

0.7
The politician should use a random process to select the sample.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Proportion is close to but not in correct range; incorrect proportion between 0 and 1 based on visible counting error; the selection of the sample is given as a reason for the difference, but randomness is not mentioned exactly.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Proportion wildly out of range, including greater than 1 or less than 0; another reasoning for the different results is given that does not account for a random process.

Teaching Notes

Accept a wide enough range of estimates for the population proportion, though it is likely that students will report the sample proportion. The large variation between the two results is the key here, recognizing that these two proportions are far away from one another. There are more formal ways to decide how much variation is considered to be significant, but those are beyond grade level.

Students are conducting a class experiment to see if there is a meaningful difference between two groups of plants that have begun to sprout leaves. The teacher randomly selects 8 plants from each group and counts the number of leaves on each plant.

Group A: 2, 2, 3, 3, 5, 5, 5, 7

Group B: 10, 9, 7, 8, 10, 12, 14, 10

Is there a meaningful difference between the two groups? Show all calculations that lead to your answer.

Answer:

Yes, there is a significant difference. Use either mean and MAD (mean absolute deviation) or median and IQR (interquartile range) to decide. Using the means, the mean of Group A is 4, and the MAD is 1.5. The mean of Group B is 10, and the MAD is 1.5. The difference in means is 6, which is 4 MADs, so it is a meaningful difference. Using the medians, the median of Group A is 4, and the IQR is 2.5. The median of Group B is 10, and the IQR is 2.5. The difference in medians is 6, which is 2.4 IQRs, so it is a meaningful difference.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: Yes, there is a meaningful difference. The median of Group A is 4. The median of Group B is 10. The IQR of each group is 2.5. The difference in medians is 2.4 IQRs.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Minor visible calculation errors cause one or two means, medians, MADs or IQRs to be incorrect. Acceptable errors: An error in calculating causes an incorrect conclusion about whether there is a meaningful difference between the two groups; an error in calculating mean or median leads to a corresponding error in calculating MAD or IQR.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Response using median and IQR does not sort Group B’s data, but otherwise correctly works through the problem; conclusion about significant differences between the groups is driven only by means or medians and not MAD or IQR; incorrect conclusion about significant differences based on correct work; mixing center and spread measures, such as using mean and IQR; incorrect calculations with no work shown but a correct conclusion based on the results.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Incorrect algorithm for calculating mean, median, MAD, or IQR; no use of these measures in problem work.

Teaching Notes

The data is built so that students can use either the mean and MAD or the median and IQR to work the problem. Either method is worth full credit, but be certain that students are consistently applying one or the other throughout.

It is also possible students come to a conclusion visually through box plots or dot plots without any of the associated calculations. Such a solution should not be worth full credit. The problem asks for calculations, and these should be worth most of the credit. If students build box plots to compare, note that the box plot for Group B has the same median and third quartile, which may confuse some students.

Scaffolding, such as asking for specific calculations, can be added to this problem, but it should not be necessary because this is a very recent topic.