Practice Set A Responses ~ Algebra/Data Analysis ~ Item 17


	Item 17 Practice Set A Responses

Amar compared the ages of the U.S. presidents who served between 1789 and 1845 at their first inaugurations to the ages of the presidents who served between 1945 and 2000 at their first inaugurations, as shown below.

Complete the following in the Answer Book:

What are the mean and median for each group of U.S. presidents?
Amar wanted to use this data to show that the ages of the U.S. presidents at their first inaugurations have decreased between 1789 and 2000. How could Amar use this data to support his conclusion?
Do you think the data provides Amar with support strong enough to make his conclusion valid? Use mathematics to justify your answer.

Score Level 3 Paper
This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. Both the means and the medians are correct, and the student correctly indicates that the means of the two sets of data could be used to support the conclusion. ("Sam could use the mean from both sets of data to show that the age of presidents has gone down.") The response correctly states that the data does not provide strong enough support to make the conclusion valid. The justification for this answer is clearly presented and fully developed. ("One reason is that the mean is less than 2 points away. Also the mean is larger in the first catagory; the median is larger in the second catagory.") This response demonstrates a complete understanding and analysis of the problem.
Score Level 2 Paper
This response indicates an incomplete application of a reasonable strategy. Both means are correct, but the medians are missing. The student correctly indicates that the means could be used to support the conclusion. ("He could support his conclusion by stating that the average age of the presidents from 1789 to 1845 is 1.3 years greater than it was from 1945 to 2000.") The response correctly states that the data does not provide strong enough support to make the conclusion valid. To justify this answer, the student notes that even though the means have decreased, the change is too small to make the conclusion valid. ("Compared to an age of around 50 to 60 years old, a 1.3 year difference between the presidents of now and then, doesn't seem that big.") This response demonstrates a conceptual understanding and analysis of the problem.
Score Level 1 Paper
This response indicates little attempt to apply a reasonable strategy. The only correct answer ("58.1") is the mean for the presidents who served between 1789 and 1845. The mean for the presidents who served between 1945 and 2000 and both medians are incorrect. No attempt is made to describe how the data could support the conclusion. Although the student correctly states that the data does not provide strong enough support to make the conclusion valid, no relevant justification is given. This response demonstrates a minimal understanding and analysis of the problem.
Score Level 2 Paper
This response indicates an incomplete application of a reasonable strategy. Both the means and the medians are correct, and the student correctly indicates that the means could be used to support the conclusion. ("He could show people the mean of the data from both sets and show that the age means went down.") Although the student correctly states that the data does not provide strong enough support to make the conclusion valid, the only justification given is that "he would need more support." Because this statement is unclear, the justification is incomplete. This response demonstrates a conceptual understanding and analysis of the problem.
Score Level 1 Paper
This response indicates little attempt to apply a reasonable strategy. The mean for the presidents who served between 1789 and 1845 is correct, but the median is incorrect. The correct answer is 57, not "56.5." Both the mean and the median for the presidents who served between 1945 and 2000 are correct. The answer for how the data could support the conclusion shows serious flaws in reasoning and does not give any relevant justification. The student incorrectly states that the data does provide support strong enough to make the conclusion valid. This response demonstrates a minimal understanding and analysis of the problem.
Score Level 3 Paper
This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. Both the means and the medians are correct. The response correctly indicates that the means could be used to support the conclusion. ("Sam could use this data to support his conclusion by comparing the means. The mean for 1789 to 1845 decreased one compared to the mean from 1845 to 2000.") The student correctly states that the data does not provide strong enough support to make the conclusion valid. Justification that is logically sound and clearly presented also is given ("I don't think this data strongly supports Sam's theory because the medians of the two time periods was about the same. One year's difference isn't that much of a span anyway.") and indicates that the difference between the ages in the two different time periods is so small that it is not strong enough to support the conclusion. This response demonstrates a complete understanding and analysis of the problem.
Score Level 1 Paper
This response indicates little attempt to apply a reasonable strategy. Both the means and the medians are correct. The student states that "He could use this data to support his conclusion by finding the mean and median for 1789 to 2000." The mean of the given data does show a decrease, but the median shows an increase and, therefore, using both the mean and median would not support the conclusion. While the student indicates that the data "might not be strong enough to support his conclusion," no justification is given to support this answer. This response demonstrates a minimal understanding and analysis of the problem.
Score Level 3 Paper
This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. Both the means and the medians are correct. The response correctly indicates that the means could be used to support the conclusion. ("If Sam wanted to show that the ages decreased he should use the mean and say that the more recent mean is lower than the older one.") Fully developed justification is given for the answer that the data does not provide strong enough support to make the conclusion valid ("only the mean is lower for the 1945-2000 presidents, and a lot of the ages are higher than the 1789-1845 presidents"). This justification clearly shows why the data is not strong enough to make the conclusion valid. This response demonstrates complete understanding and analysis of the problem.

Practice Set A Papers ~ Algebra/Data Analysis ~ Item 17

Complete the following in the Answer Book:

What are the mean and median for each group of U.S. presidents?
Amar wanted to use this data to show that the ages of the U.S. presidents at their first inaugurations have decreased between 1789 and 2000. How could Amar use this data to support his conclusion?
Do you think the data provides Amar with support strong enough to make his conclusion valid? Use mathematics to justify your answer.

Score Level 3 Paper

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. Both the means and the medians are correct, and the student correctly indicates that the means of the two sets of data could be used to support the conclusion. ("Sam could use the mean from both sets of data to show that the age of presidents has gone down.") The response correctly states that the data does not provide strong enough support to make the conclusion valid. The justification for this answer is clearly presented and fully developed. ("One reason is that the mean is less than 2 points away. Also the mean is larger in the first catagory; the median is larger in the second catagory.") This response demonstrates a complete understanding and analysis of the problem.

Score Level 2 Paper

This response indicates an incomplete application of a reasonable strategy. Both means are correct, but the medians are missing. The student correctly indicates that the means could be used to support the conclusion. ("He could support his conclusion by stating that the average age of the presidents from 1789 to 1845 is 1.3 years greater than it was from 1945 to 2000.") The response correctly states that the data does not provide strong enough support to make the conclusion valid. To justify this answer, the student notes that even though the means have decreased, the change is too small to make the conclusion valid. ("Compared to an age of around 50 to 60 years old, a 1.3 year difference between the presidents of now and then, doesn't seem that big.") This response demonstrates a conceptual understanding and analysis of the problem.

Score Level 1 Paper

This response indicates little attempt to apply a reasonable strategy. The only correct answer ("58.1") is the mean for the presidents who served between 1789 and 1845. The mean for the presidents who served between 1945 and 2000 and both medians are incorrect. No attempt is made to describe how the data could support the conclusion. Although the student correctly states that the data does not provide strong enough support to make the conclusion valid, no relevant justification is given. This response demonstrates a minimal understanding and analysis of the problem.

Score Level 2 Paper

This response indicates an incomplete application of a reasonable strategy. Both the means and the medians are correct, and the student correctly indicates that the means could be used to support the conclusion. ("He could show people the mean of the data from both sets and show that the age means went down.") Although the student correctly states that the data does not provide strong enough support to make the conclusion valid, the only justification given is that "he would need more support." Because this statement is unclear, the justification is incomplete. This response demonstrates a conceptual understanding and analysis of the problem.

Score Level 1 Paper

This response indicates little attempt to apply a reasonable strategy. The mean for the presidents who served between 1789 and 1845 is correct, but the median is incorrect. The correct answer is 57, not "56.5." Both the mean and the median for the presidents who served between 1945 and 2000 are correct. The answer for how the data could support the conclusion shows serious flaws in reasoning and does not give any relevant justification. The student incorrectly states that the data does provide support strong enough to make the conclusion valid. This response demonstrates a minimal understanding and analysis of the problem.

Score Level 3 Paper

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. Both the means and the medians are correct. The response correctly indicates that the means could be used to support the conclusion. ("Sam could use this data to support his conclusion by comparing the means. The mean for 1789 to 1845 decreased one compared to the mean from 1845 to 2000.") The student correctly states that the data does not provide strong enough support to make the conclusion valid. Justification that is logically sound and clearly presented also is given ("I don't think this data strongly supports Sam's theory because the medians of the two time periods was about the same. One year's difference isn't that much of a span anyway.") and indicates that the difference between the ages in the two different time periods is so small that it is not strong enough to support the conclusion. This response demonstrates a complete understanding and analysis of the problem.

Score Level 1 Paper

This response indicates little attempt to apply a reasonable strategy. Both the means and the medians are correct. The student states that "He could use this data to support his conclusion by finding the mean and median for 1789 to 2000." The mean of the given data does show a decrease, but the median shows an increase and, therefore, using both the mean and median would not support the conclusion. While the student indicates that the data "might not be strong enough to support his conclusion," no justification is given to support this answer. This response demonstrates a minimal understanding and analysis of the problem.

Score Level 3 Paper

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. Both the means and the medians are correct. The response correctly indicates that the means could be used to support the conclusion. ("If Sam wanted to show that the ages decreased he should use the mean and say that the more recent mean is lower than the older one.") Fully developed justification is given for the answer that the data does not provide strong enough support to make the conclusion valid ("only the mean is lower for the 1945-2000 presidents, and a lot of the ages are higher than the 1789-1845 presidents"). This justification clearly shows why the data is not strong enough to make the conclusion valid. This response demonstrates complete understanding and analysis of the problem.