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Clarifications
Clarification 1: Instruction focuses on the connection to probability.Clarification 2: Instruction includes calculating joint relative frequencies or conditional relative frequencies using tree diagrams. Clarification 3: Graphical representations include frequency tables, relative frequency tables, circle graphs and segmented bar graphs.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Categorical data
- Bivariate data
- Bar graph
- Circle graphs
- Joint relative frequency
- Conditional relative frequency
- Frequency table
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 7, students worked with categorical data to create graphical displays. In addition, students also began to develop an understanding of probabilities in grade 7, and continued this further in grade 8. In Mathematics for College Statistics, students associate the relative frequencies seen in tables and graphs with probabilities to solve real world problems.- Instruction includes reinforcing that univariate categorical data can be displayed with circle graphs and bar graphs and that the relative frequencies associated with these graphs can be viewed as probabilities.
- Instruction includes reinforcing that bivariate categorical data can be summarized with segmented bar graphs, tree diagrams, frequency two- way tables, and relative frequency two way tables. Various probabilities can be calculated from these displays.
- Instruction makes the connection to joint relative frequencies and conditional relative frequencies.
- When connecting instruction to probability, emphasize that probabilities can be represented as fractions, decimals or percentages. Students should realize that calculations from categorical data are typically probabilities/percentages/fractions since this data type is qualitative.
- Instruction includes the use of technology to create graphical displays of data.
- Instruction includes a discussion on false positives and false negatives. A false positive occurs when a test result comes back as positive when the result should have been negative. A false negative occurs when a test result comes back as negative when the result should have been positive.
Common Misconceptions or Errors
- When converting a fraction to a decimal, students may divide the denominator by the numerator.
- Students may not fully fill in a frequency two-table or a relative frequency two-way table, which would lead to incorrect calculations.
Instructional Tasks
Instructional Task 1 (MTR.7.1)- A study shows that about 2.5% of women develop the skin cancer melanoma in their lifetime. Current medical testing is not always completely accurate. In cases where a woman actually had melanoma, her test result came back positive about 94% of the time. In a case where a woman did not actually have melanoma, her test came back positive about 2% of the time. A dermatologist’s office is interested to know whether their rates are similar to the rates of the study. The results are in the two-way table below.
- Part A. Is the data being summarized univariate or bivariate?
- Part B. What is the conditional probability that a woman had a positive test given that she actually had melanoma? Is this similar to the result of the aforementioned study?
- Part C. What is the joint probability that a woman had a positive test and did not have actually have melanoma? Is this a false positive or a false negative?
- Part D. What is the conditional probability that a woman had a positive test given that she did not actually have melanoma? Is this similar to the result of the aforementioned study?
- Part E. What is the joint probability that a woman had a negative test and actually had melanoma? Is this a false positive or a false negative?
Instructional Task 2 (MTR.4.1, MTR.7.1)
- A 2014 article by Slate looked at the sex of contestants on the game show Jeopardy! and whether or not they won. The statistics provided by that article are in the tree diagram above.
- Part A. What is the probability that a woman appeared on an episode of Jeopardy! and won?
- Part B. What is the probability that a man appeared on an episode of Jeopardy! and won?
- Part C. Given that the contestant was a man, what is the probability that he won on an episode of Jeopardy!?
- Part D. Given that the contestant was a woman, what is the probability that she won on an episode of Jeopardy!?
- Part E. Discuss the content of the article and the results of Part C. and Part D. with a classmate.
Instructional Items
Instructional Item 1- Two different classes are asked their preferred format for reading books. In the first class 8 students said they preferred a hardcover book, 12 said they preferred a paperback book, 3 said they preferred an audio book and 5 said they preferred a digital format. In the second class, 5 students said they preferred a hardcover book, 14 said they preferred a paperback book, 2 said they preferred an audio book and 8 said they preferred a digital format. A segmented bar graph displays this data below.
- Part A. What is the probability that a randomly selected student from the two classes prefers a printed format (hardcover or paperback)?
- Part B. What is the probability that a randomly selected student is from class 1 and prefers an audio book?
- Part C. Given the student is from class 2, what is the probability that he or she prefers a digital format?
Related Courses
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Related Resources
Lesson Plans
Problem-Solving Tasks
Student Resources
Problem-Solving Tasks
This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.
Type: Problem-Solving Task
This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.
Type: Problem-Solving Task
Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.
Type: Problem-Solving Task
This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.
Type: Problem-Solving Task
Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.
Type: Problem-Solving Task
