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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 3.2.3 The student will communicate the use and misuse of statistics.

Assessment Limits:

  • Examples of “misuse of statistics” include the following:
    • misuse of scaling on a graph
    • misuse of measures of central tendency and variability to represent data,
    • using three-dimensional figures inappropriately
    • using data to sway interpretation to a predetermined conclusion
    • using incorrect sampling techniques
    • using data from simulations incorrectly
    • predicting well beyond the data set.

Extended Constructed Response (ECR) Item - Released in 2005

Paul asked 50 randomly selected eleventh-grade students to choose their favorite candidate for class president. The survey results are shown in the table below.

Complete the following in the Answer Book:

  • There are 300 eleventh-grade students at Paul's school. Based on the survey results, how many votes will Marie expect to receive? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
  • Using the survey results, Paul claims that Bertha will receive 2 more votes than Nick in the election. Is Paul correct? Use mathematics to justify your answer.
  • Loren asked the same question in another survey. She asked 50 randomly selected eleventh-grade chorus students to choose their favorite candidate for class president. Will you expect Loren's survey or Paul's survey to give more reliable results? Use mathematics to justify your answer.

The following 8 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Anchor Paper #1

image of student response

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The number of votes for Marie (72 votes) is correct. The explanation supports the solution, but is incomplete (taking the total votes and divided Marie's votes into it and got .24. I took .24 and multiplied it). No response is provided for the second or third parts of the question. The response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The number of votes for Marie (25) is incorrect, and no explanation is given. Although the student states that Paul is incorrect, the justification reveals a flaw in reasoning (Bertha will recieve 2 less votes than Nick will). The student believes that Paul's is the more reliable survey, and the justification supports the solution. (he is asking any eleventh grader. Loren, however, is asking just chorus students in the eleventh grade.) The response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response indicates application of an incomplete strategy. While the number of votes for Marie (90 votes) is incorrect, the explanation does reveal an appropriate strategy. The student sets up an appropriate proportion to solve for x; however, 15 is used, instead of 12. Although the student states that Paul is incorrect, the justification reveals a flaw in reasoning (more people could vote for Nick than expected). Paul's survey is chosen as the more reliable, and the justification supports the solution. (Paul is asking a mix of everyone. Lorin is only asking one group of students.) The response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response indicates application of an incomplete strategy. The number of votes for Marie (72) is correct and the explanation supports the solution (I divided 300 by 50 and got 6 then multiplied that by 12). The student states that Paul is incorrect, and the justification supports the solution (Bertha will have 12 more votes than Nick). No response is given to the third part of the problem. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to some correct solutions. The number of votes for Marie (72 votes) is correct; however, no explanation is given. The student states that Paul is incorrect, and the justification supports the solution (Bertha will get 96 votes while Nick only gets 84 votes). Paul's survey is chosen as the more reliable, and the justification supports the solution. (he survey 50 completely random 11th graders for their choise on their class president, while Loren only surveyed 50 random CHORUS students, so not everyone has a fair chance at being surveyed by Loren.) The response demonstrates a clear understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to some correct solutions. The number of votes for Marie (72 votes) is correct. The fully developed explanation supports the solution. (12/50=.24 which is the probability of Marie's vote. 300 × .24 =72.) Although the student states that Paul is incorrect, the justification reveals a flaw in reasoning (other grade student might chose the other. They only have 50% to 50%. The survey only have 50 people, Bertha might get more than 2, or Nick will be more than Bertha). Paul's survey is chosen as the more reliable, and the justification supports the solution (some students are not chorus students. Loren's idea is not equaly likely to all the students). The response demonstrates a clear understanding and analysis of the problem.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: 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 number of votes for Marie is correct. The symbolic explanation is clearly presented, fully developed, and supports the solution; the student sets up the appropriate proportion and solves for x. The student believes that Paul is incorrect. The fully developed symbolic justification supports the solution; the student sets up proportions to find the number of votes Bertha and Nick would expect, finds the difference, and notes that Nick would receive 12 votes less than Bertha. Paul's survey is chosen as the more reliable, and the justification is fully developed. (he surveyed diff. kinds of kids. Loren only surveyed a sample space of kids, chorus students.) The response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to correct solutions within the context of the problem. The number of votes for Marie is correct (72 votes). The explanation is clearly presented, fully developed and supports the solution (To get 12/50 to equal ?/300 you must multiply the top and bottom #s by 6. 12/50 • 6/6= 72/300). The student states that Paul is incorrect and the justification is fully developed; the student sets up proportions to find the number of votes Bertha and Nick would expect and then states (Nick is expected to get 84 votes and Bertha is expected to get 96). Paul's survey is chosen as the more reliable. The justification is fully developed (it is not biased toward any school groups like Loren's is toward the chorus). This response demonstrates a complete understanding and analysis of the problem.


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 3.2.3:
Skill Statements | PUBLIC RELEASE ITEMS |