2017 Ap Statistics Free Response Answers Question 6

2017 AP Statistics Free Response Answers Question 6: A full breakdown

Understanding the 2017 AP Statistics Free Response Question 6 (FRQ 6) is essential for students aiming to master the art of statistical inference and hypothesis testing. That's why this specific question focuses on the application of a one-sample z-test for a proportion, requiring students to not only perform calculations but also to justify their conclusions using a rigorous statistical framework. By analyzing the detailed answers and the logic behind the scoring rubrics, learners can develop the critical thinking skills necessary to tackle complex data analysis problems on the actual exam.

Introduction to the Problem Scenario

In the 2017 FRQ 6, the College Board presented a scenario involving a claim about a population proportion. Typically, these questions provide a context—such as a political poll, a medical study, or a consumer preference survey—and ask the student to determine if there is convincing evidence that the true proportion differs from a claimed value.

The core of this problem lies in the Hypothesis Test. To solve it, a student must deal with through four primary stages: stating the hypotheses, checking the necessary conditions for inference, calculating the test statistic and p-value, and finally, making a conclusion in the context of the problem.

Step-by-Step Breakdown of the Solution

To achieve a perfect score on a question like FRQ 6, you must follow a structured approach. Here is the detailed walkthrough of how to approach the answer.

1. Stating the Hypotheses

The first step in any inference problem is to define the parameters. You cannot simply use "p" or "x"; you must define what the proportion represents in the context of the study.

• Null Hypothesis ($H_0$): This is the statement of "no effect" or "no change." For this question, it usually takes the form $H_0: p = p_0$, where $p_0$ is the claimed proportion.
• Alternative Hypothesis ($H_a$): This is what you are trying to prove. Depending on the wording of the question ("different from," "greater than," or "less than"), this will be $p \neq p_0$, $p > p_0$, or $p < p_0$.

Pro Tip: Always use the symbol $p$ for the population proportion and $\hat{p}$ for the sample proportion. Mixing these up is a common mistake that can cost you points.

2. Checking the Conditions for Inference

Before performing the calculations, you must prove that the mathematical model you are using is valid. For a one-sample z-test for proportions, three conditions must be met:

• Randomness: The data must come from a random sample or a randomized experiment. If the problem states the sample was random, explicitly write: "The sample is stated to be random."
• Independence (10% Condition): To treat the observations as independent when sampling without replacement, the sample size ($n$) must be less than 10% of the total population. Write: "$n < 10%$ of the population."
• Normality (Large Counts Condition): The sampling distribution of $\hat{p}$ must be approximately normal. This is verified if:
  • $np_0 \geq 10$
  • $n(1 - p_0) \geq 10$ (Note: Use the null proportion $p_0$, not the sample proportion $\hat{p}$, for this check).

3. Calculating the Test Statistic and P-value

Once the conditions are verified, you move to the computation. The formula for the z-statistic is:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

After finding the z-score, you must determine the p-value. Because of that, the p-value is the probability of obtaining a result as extreme as, or more extreme than, the observed result, assuming the null hypothesis is true. * If it is a one-tailed test, find the area in one tail.

• If it is a two-tailed test ($H_a: p \neq p_0$), you must double the area of the tail.

4. The Final Conclusion

The conclusion is where many students lose points by being too vague. A complete conclusion must include:

Want to learn more? We recommend words with a soft c and writing numbers from 1 to 20 for further reading.

1. The comparison: Compare the p-value to the significance level ($\alpha$), usually 0.05.
2. The decision: State whether you "reject" or "fail to reject" the null hypothesis.
3. The context: Explain what this means regarding the original claim.

Example: "Since the p-value (0.024) is less than $\alpha = 0.05$, we reject the null hypothesis. There is convincing evidence that the true proportion of [context] is different from [claimed value]."

Scientific Explanation: Why This Method Works

The logic behind the 2017 FRQ 6 is rooted in the Central Limit Theorem (CLT). The CLT tells us that if the sample size is large enough, the sampling distribution of a proportion will be approximately normal, regardless of the shape of the population distribution.

When we calculate a z-score, we are essentially measuring how many standard deviations our sample proportion ($\hat{p}$) falls away from the hypothesized population proportion ($p_0$). That said, if the z-score is very high or very low (resulting in a tiny p-value), it suggests that the observed difference is too large to be attributed to mere sampling variability. Which means, we conclude that the null hypothesis is likely incorrect.

Common Pitfalls and How to Avoid Them

To ensure your answer aligns with the AP grading rubric, avoid these frequent errors:

• Using $\hat{p}$ in the Standard Error: A common mistake is using the sample proportion $\hat{p}$ in the denominator of the z-formula. For hypothesis tests, you must use the null proportion $p_0$.
• Vague Conclusions: Avoid saying "the result is significant." Instead, say "there is convincing evidence that..."
• Forgetting the 10% Rule: Even if it seems obvious, the graders require you to explicitly mention the 10% condition to earn the "Conditions" point.
• Incorrect Tail Logic: Ensure you check if the test is one-sided or two-sided. Forgetting to double the p-value in a two-sided test is a critical error.

FAQ: Frequently Asked Questions

Q: What happens if I don't have a calculator for the p-value? A: On the AP exam, you are expected to use a graphing calculator (like the TI-84). Use the 1-PropZTest function to find the z-statistic and p-value quickly. That said, always show the formula and your plugged-in values to get full credit.

Q: Can I use a confidence interval instead of a hypothesis test? A: Only if the question specifically asks for one. If the question asks if there is "convincing evidence," a hypothesis test is the standard and most direct way to answer.

Q: What if the Large Counts condition is not met? A: If $np_0 < 10$ or $n(1-p_0) < 10$, the sampling distribution is not normal, and a z-test cannot be reliably used. In such cases, you would need to use a Binomial Exact Test.

Conclusion

Mastering the 2017 AP Statistics FRQ 6 is about more than just getting the right number; it is about demonstrating a logical flow of statistical reasoning. By clearly stating your hypotheses, rigorously checking your conditions, accurately calculating your test statistics, and providing a context-rich conclusion, you mirror the process used by professional statisticians.

The key to success on the AP Statistics exam is consistency. Whether you are dealing with proportions or means, the framework remains the same: State $\rightarrow$ Check $\rightarrow$ Calculate $\rightarrow$ Conclude. Practice this sequence, and you will be well-equipped to handle any inference problem that comes your way.