The 2017 AP Statistics free response questions (FRQs) challenged students on a range of statistical concepts. This post provides detailed answers and explanations for each question, focusing on the key concepts and calculations required for a successful response. Remember, while these explanations provide a strong framework, your own responses should demonstrate a complete understanding and show your work clearly.
Question 1: Inference for a Population Proportion
This question involved analyzing data related to the proportion of people who support a certain policy.
Key Concepts:
- One-sample z-test for a proportion: This is the core statistical method needed to answer this question. You'll need to calculate the sample proportion, the standard error, the z-statistic, and the p-value.
- Conditions for inference: Before conducting the test, you must verify the conditions for inference: Random sample, independence (10% condition), and large sample size (success-failure condition: np ≥ 10 and n(1-p) ≥ 10).
- Interpreting the p-value: This determines whether you reject or fail to reject the null hypothesis. A small p-value (typically less than 0.05) provides evidence against the null hypothesis.
- Confidence intervals: You might also be asked to construct a confidence interval for the population proportion. This provides a range of plausible values for the true proportion.
Answer Outline:
- State: Clearly state the null and alternative hypotheses.
- Plan: Verify the conditions for inference. Specify the test you're using (one-sample z-test for a proportion).
- Do: Calculate the sample proportion, the standard error, the z-statistic, and the p-value. Show your work clearly!
- Conclude: Interpret the p-value in the context of the problem. State your conclusion about the null hypothesis and draw a conclusion in the context of the problem. If constructing a confidence interval, interpret the interval in context.
Important Considerations:
- Pay close attention to the wording of the question. Are you performing a one-sided or two-sided test?
- Always show your work. Points are awarded for correct calculations and justifications.
- Clearly state your conclusion in the context of the problem.
Question 2: Inference for Two Population Means
This question likely focused on comparing the means of two populations.
Key Concepts:
- Two-sample t-test: This is the most probable method needed here. This test compares the means of two independent groups.
- Pooled vs. unpooled t-test: Decide whether to pool the variances based on whether you can assume equal variances in the two populations. A test for equality of variances (like an F-test) might be helpful here.
- Conditions for inference: Again, ensure you verify the conditions for inference, including random sampling, independence, and approximate normality of the sampling distributions (or large sample sizes).
- Interpreting results: Analyze the t-statistic and p-value to reach a conclusion about the difference between the population means.
Answer Outline (similar to Question 1):
- State: Clearly state the null and alternative hypotheses.
- Plan: Verify the conditions for inference. Specify the test you're using (two-sample t-test, stating whether you pooled the variances).
- Do: Perform the calculations necessary to obtain the t-statistic and p-value.
- Conclude: Interpret your findings in context.
Question 3: Inference for Regression
This question likely involved analyzing a linear regression model.
Key Concepts:
- Linear Regression Model: Understand the assumptions underlying linear regression (linearity, independence, normality of residuals, constant variance).
- Hypothesis testing for slope: Test whether the slope of the regression line is significantly different from zero. This helps determine if there is a statistically significant linear relationship between the variables.
- Confidence intervals for slope: Construct a confidence interval for the slope to estimate the range of plausible values for the true slope.
- Interpreting R-squared: This tells you the proportion of variance in the response variable that is explained by the explanatory variable.
Answer Outline:
- State: State the hypotheses for the slope of the regression line.
- Plan: Verify the conditions for inference in linear regression. State the test you will use (t-test for the slope).
- Do: Perform the calculations needed to find the t-statistic, p-value, and confidence interval. Show your work.
- Conclude: Interpret your findings regarding the relationship between the variables and the strength of the relationship. Comment on R-squared.
General Tips for all FRQs:
- Clearly define your variables. Make sure it is clear what each variable represents.
- Show your work meticulously. Points are often awarded for demonstrating the steps in your calculations.
- Explain your reasoning clearly. Don't just state your answers; justify them.
- Use proper statistical notation. Use symbols correctly.
- Context is crucial. Make sure to interpret your results in the context of the problem.
Remember to consult the official AP Statistics scoring guidelines for the specific 2017 exam to check your answers against the rubric. Good luck!
