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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 2003

Juan conducted two different surveys about student transportation to school. For Survey A, he stood at the entrance of the school at 7:30 a.m. and surveyed the first 80 students. For Survey B, he used a random number generator to select 80 students in the school to survey. His results are shown in the tables below.

Complete the following in the Answer Book:

  • Use principles of simple random sampling to justify why Juan should have more confidence in the results of Survey B.
  • Use principles of simple random sampling to justify why Juan should have less confidence in the results of Survey A.
  • According to the data collected from Survey B, of the 400 total students in the school, how many students should Juan expect to ride a bike? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

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 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The student provides a fully developed justification for more confidence in Survey B. ("Everyone has a fair chance, and every person is chosen independently.") The justification for less confidence in Survey A is fully developed and clearly presented. ("Some students could get to school late and that means it is not a fair chance for everyone to get picked.") The student provides the correct number of expected bike-riding students, and the explanation is fully developed. ("I multiplied 400 {# of students} times 10 {# of students}, then I divided 4,000 by 80 {people surveyed}.) This response demonstrates a complete 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 attempted justification for more confidence in Survey B, and less in Survey A, does not incorporate equal chance or independent selection. Used repeatedly, the term "random" is given in the question and, therefore, needs further clarification. The correct number of expected bike-riding students is provided. By setting up a proportion and solving for x, a symbolic explanation is given which supports the solution. This response demonstrates a minimal understanding and analysis of the problem. Compare to Anchor Paper #2.
Sample Student Response #3

image of student response

Score for Sample Student Response #3: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to some correct solutions. The student's justification for more confidence in Survey B is generally well developed ("gives an equal chance of all the students to be represented") and addresses equal chance, but not independent selection. The justification for less confidence in Survey A supports the solution ("looking at the first 80 people takes away the chance for all the people who come to school later"). The number of expected bike-riding students is correct, and the symbolic explanation supports the solution. This response demonstrates a clear understanding and analysis of the problem. Compare to Anchor Paper #6.
Sample Student Response #4

image of student response

Score for Sample Student Response #4: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student's justification for more confidence in Survey B supports the solution. ("It gives every person an equal chance and randomly selected the 80 people that were chosen for the survey.") This justification does not address independent selection. The justification for less confidence in Survey A supports the solution ("…it is very exclusionary. Any students who arrive later had no chance of being selected to be surveyed"). The answer to the third part of the question is incomplete. This response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #3.

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
/share/clg/xml/public_release/mathematics/2003_323_alg21.xml
Resources for 3.2.3:
Skill Statements | PUBLIC RELEASE ITEMS |