End-of-Unit Assessment

1.

Elena would like to know the average height of seventh graders in her school district. She measures the heights of everyone in a random sample of 20 students. The mean height of Elena’s sample is 61 inches, and the MAD (mean absolute deviation) is 2 inches.

Select all the true statements.

A.

The median height of the sample must be between 59 and 63 inches.

B.

Another random sample of 20 students is likely to have a mean between 57 and 65 inches.

C.

The mean height of these 20 students is likely to be exactly the same as the mean height of all students in the district.

D.

The mean height of these 20 students is likely to be exactly the same as the mean height of a second random sample of 20 students.

E.

Elena would be more likely to get an accurate estimate of the mean height of the population by sampling 40 people instead of sampling 20 people.

Answer: B, E

Teaching Notes

Students who select choice A are incorrectly using the MAD and mean information applied to the median. Students who do not select choice B have not learned this unit’s concept about twice the MAD determining expected variation. Students who select choice C 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 E 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.

2.

Here is a dot plot showing how much time customers spent in a store, rounded to the nearest five minutes.

Dot plot from 0 to 50. Time in store in minutes. Beginning at 0, number of dots above each increment is 0, 17, 4, 3, 4, 8, 9, 6, 2, 0, 0.

Which of the following is a representative sample of this population?

A dot plot, A, from 0 to 50. Time in store in minutes. — A

<p>A dot plot, B, from 0 to 50. Time in store in minutes.</p> — B

A dot plot, C, from 0 to 50. Time in store in minutes. — C

A dot plot, D, from 0 to 50. Time in store in minutes. — D

A.

A

B.

B

C.

C

D.

D

Answer:

D

Teaching Notes

Students who select choice A noticed the overall shape of the distribution but did not take into account the large peak of data at 5 minutes. Students who select choice B noticed the overall shape of the distribution but did not notice a shift in the data. Students who select choice C may believe that samples must have a symmetric distribution, which is not true.

3.

The heights for a random sample of 50 volleyball players across the United States are summarized in this histogram.

<p>Histogram from 66 to 80 by 2’s. Height in inches. Beginning at 66 up to but not including 68, height of bar at each interval is 2,3,13,17,7,5,1.</p><br>

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

A.

The median height for volleyball players on a single team is about 73 inches.

B.

The median height for volleyball players in the United States is about 73 inches.

C.

There are no volleyball players in the United States taller than 80 inches.

D.

The mean height for volleyball players in the United States is about 73 inches.

E.

The mean height for basketball players in the United States is about 73 inches.

Answer: B, D

Teaching Notes

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

4.

An administrator of a large middle school is installing some vending machines in the cafeteria and teacher’s lounge. She wants to know what type of machine would be most popular.

What is the population for the administrator’s question?
Give an example of a sample the administrator could use to help answer her question that is likely to be representative.

Answer:

Sample response:

The population is current students, teachers, and staff of the school.
Use a list of students, teachers, and staff, and select 30 of them 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 regularly spend time in the school, not just students. There is lots of flexibility in acceptable responses to the second question since the item doesn't say how large the school is, but it should mention using a random process.

5.

All the students at 2 different schools take the same assessment. The principal at each school uses the scores from one class at their school to prepare information to share. Here is what they brought.

School A:

Mean: 68.8
Median: 72
MAD: 8.02
IQR: 20

School B:

Box plot from 55 to 100 by 5’s. Whisker from 60 to 70. Box from 70 to 80 with vertical line at 75. Whisker from 80 to 90.

Which school has better typical results on the exam?
Which school has greater variability in the results?

Answer:

School B
School A

Teaching Notes

Some students choose to make a box plot for School A. 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.

6.

A box by the exit of a store has a sign asking, “Were you happy with your experience today?” Customers may write yes or no on a slip of paper and leave it in the box. The store owner stops 12 people on their way out and asks them the same question. These are the results:

yes, yes, yes, no, yes, no, yes, yes, no, yes, no, yes

Based on these 12 responses, estimate the proportion of all shoppers who are happy with their experience that day.
After collecting the papers from the box, it shows that only 25% of the slips are marked yes. What might explain the difference between the actual results and the owner’s survey? How could the owner have done the survey better?

Answer:

Sample response:

$\frac{8}{12}$ or $\frac{2}{3}$ . (Any answer between 0.6 and 0.75 is reasonable.)
The sample was not selected using a random process. The owner should have tried to select people for the survey using more randomness to reduce bias.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$\frac{8}{12}$
The survey wasn’t random. Use more randomness to get 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; 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.

7.

A scientist wants to know if there is a meaningful difference between two groups of gels that grow bacteria. He randomly selects six gels from each group, and counts the number of bacteria spots on each gel:

Group A: 9, 12, 13, 13, 14, 17

Group B: 8, 6, 5, 8, 13, 8

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 13, and the MAD is 1.67. The mean of Group B is 8, and the MAD is 1.67. The difference in means is 3 MADs, so it is a meaningful difference. Using the medians, the median of Group A is 13, and the IQR is 2. The median of Group B is 8, and the IQR is 2. The difference in medians is 2.5 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 13. The median of Group B is 8. The IQR of each group is 2. The difference in medians is more than 2 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.