AP Statistics 8.7 Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data Study Notes
AP Statistics 8.7 Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data Study Notes- New syllabus
AP Statistics 8.7 Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- The chi-square distribution may be used to model variation.
Key Concepts:
- Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data
Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data
Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data
We use this to determine which statistical procedure to use for categorical data, depending on the research question, number of populations, and number of variables.
Guidelines for Selection:
One categorical variable:
- 1-sample proportion (z) inference:
- Estimating a population proportion → 1-sample confidence interval for a proportion
- Testing a population proportion → 1-sample z-test for a proportion
- Goodness-of-fit (χ²) test: – Comparing observed counts to expected counts for more than two categories
Two categorical variables:
- Chi-square test for independence: – Determining whether two variables are associated in a single population
- Chi-square test for homogeneity: – Comparing distributions of a categorical variable across two or more populations
Notes:
- Always check that conditions for the chosen procedure are met (randomness, independence, expected counts ≥ 5).
- Hypotheses must refer to population proportions or distributions.
- Choose a confidence level (\(C\)) or significance level (\(\alpha\)) appropriate for the context.
Example
A researcher wants to determine whether a new teaching method affects students’ preferred study location (Library, Home, Cafe). 50 students are surveyed.
Which statistical inference procedure is appropriate?
▶️ Answer / Explanation
Step 1 — Identify variables:
One categorical variable: study location (3 categories)
Step 2 — Determine type of inference:
- Since there are more than 2 categories and the researcher wants to compare observed counts to expected counts → Chi-Square Goodness-of-Fit Test
Step 3 — Verify conditions:
- Random sample of students
- Expected counts ≥ 5 in each category
- Observations independent
Conclusion: Use a chi-square goodness-of-fit test to determine if the distribution of preferred study locations differs from the expected distribution.
Example
A company claims that the distribution of its product colors sold is 40% Red, 35% Blue, and 25% Green. A sample of 100 products sold is collected.
Which statistical inference procedure should be used to test if the observed color distribution differs from the claimed distribution?
▶️ Answer / Explanation
Step 1 — Identify variable: Color (Red, Blue, Green) — one categorical variable
Step 2 — Determine procedure: – Comparing observed counts to expected counts for multiple categories → Chi-Square Goodness-of-Fit Test
Step 3 — Conditions:
- Random sample
- Expected counts ≥ 5
- Observations independent
Conclusion: Use a chi-square goodness-of-fit test to assess whether the color distribution of products sold matches the claimed distribution.
Example
A researcher surveys 150 students from three different schools (School A, B, C) about whether they participate in after-school sports (Yes, No).
Which statistical inference procedure should be used to compare participation rates among the three schools?
▶️ Answer / Explanation
Step 1 — Identify variables: – School (A, B, C) — categorical – Participation (Yes, No) — categorical
Step 2 — Determine procedure: – Comparing the distribution of participation across multiple populations → Chi-Square Test for Homogeneity
Step 3 — Conditions:
- Random samples from each school
- Expected counts in each cell ≥ 5
- Observations independent
Conclusion: Use a chi-square test for homogeneity to determine whether participation rates differ among the three schools.