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Goal 3 Data Analysis And Probability

Expectation 3.2 The student will apply the basic concepts of statistics and probability to predict possible outcomes of real-world situations.

Indicator 3.2.1 The student will make informed decisions and predictions based upon the results of simulations and data from research.

Extended Constructed Response (ECR) Item - Released in 2001

A movie producer is considering two different endings for a movie. To decide which ending is better, the producer randomly selected two groups of people to watch each ending. There were 200 people in each group. The two groups rated the movie endings on a scale of 1 to 100. The box-and-whisker plots below show a summary of their results.

Complete the following in the Answer Book:

  • Which ending had a higher median rating? Use mathematics to justify your answer.
  • Which ending had a wider range of ratings? Use mathematics to justify your answer.
  • Based on the data shown in the box-and-whisker plots, the movie producer decided that the first ending was better than the second ending. Is this a valid conclusion to make from the data given? Use mathematics to justify your answer.

The following 4 Sample Student Responses represent a range of score points.

Sample Student Response #1

image of student response

Score for Sample Student Response #1: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student's analysis of the box-and-whisker plot correctly identifies the second ending as having the higher median. The justification for this solution is complete ("...the 1st ending was 72 and the 2nd was 78."). The student also correctly identifies the first ending as having the wider range and provides correct justification ("...the Range of the 2nd ending was 38 and the 1st was 42"). However, there is no indication that the student has applied a reasonable strategy to address whether the producer made a valid conclusion. This response demonstrates only a conceptual understanding and analysis of the problem.


Sample Student Response #2

image of student response

Score for Sample Student Response #2: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The student's analysis of the box-and-whisker plot correctly identifies the second ending as having the higher median. The student's choice of the first ending as the one with the wider range is also correct. However, there is no justification provided for either solution. In addition, the student does not provide any appropriate justification for disagreeing with the producer's conclusion. This response demonstrates a minimal understanding and analysis of the problem.


Sample Student Response #3

image of student response

Score for Sample Student Response #3: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The student correctly identifies the second ending as having the higher median. The justification for this solution is complete ("The rating was 78 and the first ending was 72."). The student also correctly identifies the first ending as having the wider range and provides a complete justification ("The first endings range was from 53-96. The second ending ranged from 52-90.") The student justifies the second ending as the better one by interpreting the significance of a higher median ("The median was higher for the second ending. This means it got higher scores...Half the ratings were above 78."). This justification is logically sound, clearly presented, and supports the solution. This response demonstrates a complete understanding and analysis of the problem.


Sample Student Response #4

image of student response

Score for Sample Student Response #4: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution. The student correctly identifies the second ending as having the higher median and provides a complete justification ("The first endings median was 72 and the second was 78."). The student's correct choice for the first ending as having the wider range is also fully justified ("...it's range was from 54-96 + the second ending had a range from 52-90"). The student attempts to justify the second ending as the better one by comparing the majorities ("In the first ending the majority of the ratings were from 62-72 while in the second ending the majority was from 66-84."). Although this assessment is plausible, the student's use of the term "majority," rather than "half," indicates an incorrect interpretation of the box-and-whisker plot. This response demonstrates a clear understanding and analysis of the problem.


Additional Resources

Anchor Papers used in scoring

Extended Constructed Response (ECR) Rubric
Print: Scoring Rubric (pdf)
Score 4

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

Score 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Score 2

The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Score 1

The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Score 0

The response is completely incorrect or irrelevant. There may be no response, or the response may state, “I don't know.”

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Last Revised 8/16/00
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Resources for 3.2.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |