Item 17 Practice Set B 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 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. All four measures of central tendency are correct. The response correctly identifies the mean as supporting the conclusion. ("He could use the mean to show that the ages have decreased because the mean from 1789-1845 is greater than the mean from 1945-2000.") The student incorrectly tries to justify that the data is strong enough to make the conclusion valid. ("Yes I do think he does have strong enough data because the mean clearly shows that the ages have decreased.") Even though the means do show a decrease, the medians show an increase and, therefore, would not offer support strong enough to make the conclusion valid. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

Score Level 1 Paper

 

This response indicates little attempt to apply a reasonable strategy. The means for both sets of data and the median for the presidents who served between 1945 and 2000 are correct. However, the median for the presidents who served between 1789 and 1845 is incorrect ("57.5" instead of 57). No attempt is made to answer the remainder of the question. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

Score Level 3 Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. All four measures of central tendency are correct. The student correctly indicates that the means could be used to support the conclusion ("use the mean because it goes down by 1.3 years"). The response correctly states that the data does not provide strong enough support to make the conclusion valid. The justification given to support this answer is clearly presented. ("I don't think the data provides enough strong data to support his conclusion because the mean may go down but the median {middle age} goes up by one year, therefore causing a disagreement over if it's up or down.")

image of student response

Score Level 1 Paper

 

This response indicates little attempt to apply a reasonable strategy. All four measures of central tendency are correct; however, the response to the second bullet does not describe how data could be used to support the conclusion. The student incorrectly states that the data is strong enough to make the conclusion valid. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. Both the mean and the median are correct for the presidents who served between 1789 and 1845. The mean is correct for the presidents who served between 1945 and 2000, but the median is incorrect ("60.5" instead of 58). The student correctly states that the mean could be used to support the conclusion and clearly presents justification to show that the data does not provide support strong enough to make the conclusion valid ("even if he used the means the difference between them is less than two years"). This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

Score Level 3 Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. All four measures of central tendency are correct. The student clearly indicates that in order for Sam to support his conclusion he should use the means. ("Sam could use the data to support his conclusion by using the mean of both Presidents' inaugurations, since the first term was 58.1, and the 2nd term was 56.8.") Logically sound and fully developed justification is presented to show that the data does not offer support strong enough to make Sam's conclusion valid. ("He's trying to say that the ages decreased, so he couldn't use the median, because that increases. I don't think it is a strong point to use though, because it only decreases by about 1.3 years. So it shows that it decreased, but not by much, so it might not be valid enough.") This response demonstrates a complete understanding and analysis of the problem.

image of student response

Score Level 1 Paper

 

This response indicates little attempt to apply a reasonable strategy. All four measures of central tendency are correct; however, the student's comparison of the minimum and maximum ages of the two groups does not support the conclusion. The response incorrectly states that the data provides support strong enough to make the conclusion valid. Comparing the age ranges of the two sets of data cannot justify that the data is strong enough to make the conclusion valid. This response demonstrates a minimal understanding of the problem.

image of student response

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 student correctly indicates that the means could be used to support the conclusion ("he should use the mean of the ages because median shows increase"). The student correctly states that the data does not provide strong enough support to make the conclusion valid. Justification that supports the solution and is logically sound also is given. "It needs more back up than just the means of the ages," and, therefore, the data would not be strong enough to make the conclusion valid. This response demonstrates a complete understanding and analysis of the problem.

image of student response

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. All four measures of central tendency are correct. The response correctly identifies the mean as supporting the conclusion ("he could use the mean. It went from 58.1 to 56.8"). The student incorrectly states that the data is strong enough to make the conclusion valid. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response
 

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

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 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. All four measures of central tendency are correct. The response correctly identifies the mean as supporting the conclusion. ("He could use the mean to show that the ages have decreased because the mean from 1789-1845 is greater than the mean from 1945-2000.") The student incorrectly tries to justify that the data is strong enough to make the conclusion valid. ("Yes I do think he does have strong enough data because the mean clearly shows that the ages have decreased.") Even though the means do show a decrease, the medians show an increase and, therefore, would not offer support strong enough to make the conclusion valid. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

 

Score Level 1 Paper

 

This response indicates little attempt to apply a reasonable strategy. The means for both sets of data and the median for the presidents who served between 1945 and 2000 are correct. However, the median for the presidents who served between 1789 and 1845 is incorrect ("57.5" instead of 57). No attempt is made to answer the remainder of the question. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

 

Score Level 3 Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. All four measures of central tendency are correct. The student correctly indicates that the means could be used to support the conclusion ("use the mean because it goes down by 1.3 years"). The response correctly states that the data does not provide strong enough support to make the conclusion valid. The justification given to support this answer is clearly presented. ("I don't think the data provides enough strong data to support his conclusion because the mean may go down but the median {middle age} goes up by one year, therefore causing a disagreement over if it's up or down.")

image of student response

 

Score Level 1 Paper

 

This response indicates little attempt to apply a reasonable strategy. All four measures of central tendency are correct; however, the response to the second bullet does not describe how data could be used to support the conclusion. The student incorrectly states that the data is strong enough to make the conclusion valid. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

 

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. Both the mean and the median are correct for the presidents who served between 1789 and 1845. The mean is correct for the presidents who served between 1945 and 2000, but the median is incorrect ("60.5" instead of 58). The student correctly states that the mean could be used to support the conclusion and clearly presents justification to show that the data does not provide support strong enough to make the conclusion valid ("even if he used the means the difference between them is less than two years"). This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

 

Score Level 3 Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. All four measures of central tendency are correct. The student clearly indicates that in order for Sam to support his conclusion he should use the means. ("Sam could use the data to support his conclusion by using the mean of both Presidents' inaugurations, since the first term was 58.1, and the 2nd term was 56.8.") Logically sound and fully developed justification is presented to show that the data does not offer support strong enough to make Sam's conclusion valid. ("He's trying to say that the ages decreased, so he couldn't use the median, because that increases. I don't think it is a strong point to use though, because it only decreases by about 1.3 years. So it shows that it decreased, but not by much, so it might not be valid enough.") This response demonstrates a complete understanding and analysis of the problem.

image of student response

 

Score Level 1 Paper

 

This response indicates little attempt to apply a reasonable strategy. All four measures of central tendency are correct; however, the student's comparison of the minimum and maximum ages of the two groups does not support the conclusion. The response incorrectly states that the data provides support strong enough to make the conclusion valid. Comparing the age ranges of the two sets of data cannot justify that the data is strong enough to make the conclusion valid. This response demonstrates a minimal understanding of the problem.

image of student response

 

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 student correctly indicates that the means could be used to support the conclusion ("he should use the mean of the ages because median shows increase"). The student correctly states that the data does not provide strong enough support to make the conclusion valid. Justification that supports the solution and is logically sound also is given. "It needs more back up than just the means of the ages," and, therefore, the data would not be strong enough to make the conclusion valid. This response demonstrates a complete understanding and analysis of the problem.

image of student response

 

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. All four measures of central tendency are correct. The response correctly identifies the mean as supporting the conclusion ("he could use the mean. It went from 58.1 to 56.8"). The student incorrectly states that the data is strong enough to make the conclusion valid. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response