AP Statistics 9.6 Skills Focus: Selecting an Appropriate Inference Procedure Study Notes
AP Statistics 9.6 Skills Focus: Selecting an Appropriate Inference Procedure Study Notes- New syllabus
AP Statistics 9.6 Skills Focus: Selecting an Appropriate Inference Procedure Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- Selecting an Appropriate Inference Procedure
Key Concepts:
- Skills Focus: Selecting an Appropriate Inference Procedure
Skills Focus: Selecting an Appropriate Inference Procedure
Skills Focus: Selecting an Appropriate Inference Procedure
Choosing the correct inference procedure depends on the type of variable, the research question, and the study design. Here’s a structured approach:
1. Categorical Response Variable (Proportions)
- One-sample proportion: Use a 1-sample z-test or 1-sample z-interval for population proportion.
- Two-sample proportions: Use a 2-sample z-test or 2-sample z-interval for the difference between proportions.
- Goodness-of-fit / homogeneity / independence: Use chi-square tests.
2. Quantitative Response Variable (Means or Slopes)
- One-sample mean: Use a 1-sample t-test or t-interval for a population mean.
- Two–sample means: Use a 2-sample t-test or t-interval for difference of means.
- Regression slope: Use a t-test or t-interval for the slope (\(\beta\)) of a regression model.
3. Paired Data (Quantitative Differences)
- Paired t-test / t-interval for mean difference (\(\mu_d\)) when observations are dependent.
Notes:
- Always check assumptions before performing the inference procedure (e.g., normality, independence, sample size conditions).
- The choice between a confidence interval and a hypothesis test depends on whether the goal is estimation or testing a claim.
- For multiple populations or complex designs, consider ANOVA or multiple regression as appropriate.
Example
A survey of 120 students finds 78 prefer online classes. Construct a 95% confidence interval for the true proportion of students who prefer online classes.
▶️ Answer / Explanation
Step 1 — Identify the formula: \(\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\)
Step 2 — Sample proportion: \(\hat{p} = \dfrac{78}{120} = 0.65\)
Step 3 — Critical value for 95%: \(z^* = 1.96\)
Step 4 — Standard error: \(\sqrt{\dfrac{0.65(1-0.65)}{120}} = 0.044\)
Step 5 — Margin of error: \(1.96 \times 0.044 = 0.086\)
Step 6 — Confidence interval: \(0.65 \pm 0.086 = (0.564, 0.736)\)
Example
Class A (n=25, mean=85, s=4) and Class B (n=30, mean=80, s=5). Test at \(\alpha = 0.05\) if the average scores differ.
▶️ Answer / Explanation
Step 1 — Identify test: 2-sample t-test
Step 2 — Test statistic: \(t = \dfrac{\bar{x}_1 – \bar{x}_2}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}} = \dfrac{85 – 80}{\sqrt{\dfrac{16}{25} + \dfrac{25}{30}}} = \dfrac{5}{\sqrt{0.64 + 0.8333}} = \dfrac{5}{1.206} \approx 4.15\)
Step 3 — df ≈ 50 (approximation)
Step 4 — p-value < 0.001, reject \(H_0\). Conclusion: significant difference in class scores.
Example
A study collects data on weekly study hours (\(x\)) and exam scores (\(y\)) for 15 students. Regression slope \(b = 2.3\), \(SE_b = 0.9\). Test if slope differs from 0 at \(\alpha = 0.05\).
▶️ Answer / Explanation
Step 1 — Test statistic: \(t = \dfrac{b – 0}{SE_b} = \dfrac{2.3}{0.9} \approx 2.56\)
Step 2 — df = n – 2 = 13
Step 3 — p-value ≈ 0.024 < 0.05, reject \(H_0\)
Step 4 — Conclusion: There is significant evidence that study hours positively affect exam scores.
Example
A candy company claims equal sales for 3 flavors: chocolate, vanilla, strawberry. A sample of 150 sales shows 60, 50, and 40, respectively. Test at \(\alpha = 0.05\) if sales are equally distributed.
▶️ Answer / Explanation
Step 1 — Expected counts: 150 ÷ 3 = 50 each
Step 2 — Chi-square statistic: \(\chi^2 = \sum \dfrac{(O-E)^2}{E} = \dfrac{(60-50)^2}{50} + \dfrac{(50-50)^2}{50} + \dfrac{(40-50)^2}{50} = 2 + 0 + 2 = 4\)
Step 3 — df = k-1 = 3-1 = 2
Step 4 — Critical value \(\chi^2_{0.05,2} = 5.991\), 4 < 5.991 → fail to reject \(H_0\)
Conclusion: No significant evidence sales differ from equal proportions.