#AP Statistics: Confidence Intervals & Hypothesis Testing for Two Proportions 🚀

Hey there, future AP Stats master! Let's break down confidence intervals and hypothesis testing for two proportions. You've got this! 💪

#Confidence Intervals for the Difference of Two Proportions

#Understanding the Basics

When we're comparing two groups, we often want to know if there's a real difference in their proportions. For example, is the proportion of people who prefer Brand A different from the proportion who prefer Brand B? Confidence intervals help us estimate this difference.

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• A confidence interval gives us a range of plausible values for the true difference between two population proportions.
• It's crucial to remember that we're making inferences about population proportions, not just the sample proportions we calculated.
• The confidence level (usually 95%) tells us how confident we are that our interval captures the true difference. For example, a 95% confidence level means that if we repeated the sampling process many times, 95% of the resulting intervals would contain the true difference.

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When you write out your interpretation, make sure you include:

1. Confidence Level: State the confidence level given in the problem (e.g., "We are 95% confident...").
2. Population Proportions: Explicitly mention that you're making inferences about the difference in population proportions, not sample proportions.
3. Context: Relate your interpretation to the specific problem. What are the two groups you're comparing? What are you measuring?

#Example Interpretation

Let’s say we're comparing the proportion of students who prefer online learning to the proportion who prefer in-person learning. Our 95% confidence interval for the difference in proportions is (0.10, 0.25). A good interpretation would be:

"We are 95% confident that the true difference in the proportion of students who prefer online learning and the proportion who prefer in-person learning is between 0.10 and 0.25. This means that the proportion of students who prefer online learning is likely between 10% and 25% higher than the proportion who prefer in-person learning."

#Hypothesis Testing with Confidence Intervals

#Using Confidence Intervals to Test Claims

Confidence intervals aren't just for estimation; they're also super useful for testing claims! Here’s the secret: 🤫

• The Key Value: Zero 🔑: When testing for a difference in proportions, we're often interested in whether the difference could be zero. If the difference is zero, there is no difference between the two population proportions.
• Zero Inside the Interval? If our confidence interval includes zero, it means that a difference of zero is a plausible value. Therefore, we cannot reject the null hypothesis (that there's no difference).
• Zero Outside the Interval? If zero is not included in the interval, it suggests that the true difference is likely not zero, and we can reject the null hypothesis in favor of the alternative hypothesis (that there is a difference).

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Think of zero as a gatekeeper! 🚪

• If the gatekeeper (zero) is inside the interval, the null hypothesis gets to stay (we fail to reject it).
• If the gatekeeper (zero) is outside the interval, the null hypothesis has to leave (we reject it).

#Example: Michael Jordan vs. Lebron James 🏀

Let's revisit the example of comparing Michael Jordan and Lebron James's shooting percentages. Suppose our 95% confidence interval for the difference in their population proportions of shots made is (0.063, 0.133).

• Interpretation: "We are 95% confident that the true difference in the population proportions for shots made between Michael Jordan and Lebron James is between 0.063 and 0.133."
• Hypothesis Test: Since 0 is not included in the interval, we have reasonable evidence to suggest that there is a difference in their shooting percentages. We can reject the null hypothesis that there is no difference.

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• Confounding Variables: Always think about other factors that might influence the results. In our basketball example, things like teammates and the season of their career could be confounding variables.
• Sample Representativeness: Make sure your samples are representative of the populations you're studying. Using only one season of data might not give you the full picture.

#Final Exam Focus

#High-Priority Topics

• Interpreting Confidence Intervals: Focus on including the confidence level, population parameter, and context.
• Connecting Intervals to Hypothesis Tests: Understand how the inclusion or exclusion of zero affects your conclusion.
• Identifying Confounding Variables: Be ready to discuss how other factors might influence the results.

#Common Question Types

• Multiple Choice: Expect questions that test your understanding of confidence interval interpretation and the relationship between intervals and hypothesis tests.
• Free Response: Be prepared to construct and interpret confidence intervals, and use them to test claims. Always provide context.

#Last-Minute Tips

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
• Common Pitfalls: Avoid confusing sample proportions with population proportions. Always interpret in context.
• Strategies: Read each question carefully and identify the key concepts being tested. Show all your work, even if you're using a calculator.

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Practice Question

Practice Questions

#Multiple Choice Questions

1. A researcher wants to compare the proportion of students who prefer coffee to the proportion of students who prefer tea. They take a random sample of students and construct a 95% confidence interval for the difference in proportions. The resulting interval is (-0.05, 0.15). Which of the following is a correct conclusion? (A) There is a significant difference in the proportions of students who prefer coffee and tea. (B) There is no significant difference in the proportions of students who prefer coffee and tea. (C) There is a 95% probability that the true difference in proportions is between -0.05 and 0.15. (D) We are 95% confident that the sample proportion difference is between -0.05 and 0.15. 2. A 90% confidence interval for the difference in two population proportions is (0.02, 0.10). Based on this interval, which of the following is a correct conclusion? (A) The two population proportions are equal. (B) The two population proportions are likely different. (C) There is a 90% probability that the true difference in proportions is between 0.02 and 0.10. (D) We are 90% confident that the sample proportion difference is between 0.02 and 0.10. 3. A study compared the proportion of adults who exercise regularly in two different cities. A 99% confidence interval for the difference in proportions was found to be (-0.08, 0.03). Which of the following is a correct interpretation of this interval? (A) We are 99% confident that the true difference in the proportions of adults who exercise regularly in the two cities is between -0.08 and 0.03. (B) There is a 99% chance that the true difference in proportions is between -0.08 and 0.03. (C) The sample proportions are different in the two cities. (D) The population proportions are different in the two cities.

#Free Response Question

A researcher is investigating the effectiveness of a new medication for reducing blood pressure. They randomly assign 100 patients with high blood pressure to either a treatment group (receiving the new medication) or a control group (receiving a placebo). After one month, they measure the proportion of patients in each group whose blood pressure has decreased.

The results are as follows:

• Treatment Group: 45 out of 100 patients showed a decrease in blood pressure.
• Control Group: 30 out of 100 patients showed a decrease in blood pressure.

(a) Construct a 95% confidence interval for the difference in the proportions of patients showing a decrease in blood pressure between the treatment and control groups. Show all your work.

(b) Interpret the confidence interval in the context of the study.

(c) Based on the confidence interval, is there evidence to suggest that the new medication is effective in reducing blood pressure? Explain.

#FRQ Scoring Rubric

(a) Constructing the Confidence Interval (4 points)

• 1 point: Correctly calculating the sample proportions:
  • Treatment group: $\hat{p}_1 = 45/100 = 0.45$
  • Control group: $\hat{p}_2 = 30/100 = 0.30$
• 1 point: Correctly calculating the standard error: $$SE = \sqrt{\frac{0.45(1-0.45)}{100} + \frac{0.30(1-0.30)}{100}} ≈ 0.066$$
• 1 point: Correctly finding the critical value for a 95% confidence interval (z* = 1.96)
• 1 point: Correctly calculating the confidence interval: $(0.45 - 0.30) ± 1.96 * 0.066 = 0.15 ± 0.129 ≈ (0.021, 0.279)$

(b) Interpreting the Confidence Interval (2 points)

• 1 point: Correctly stating the confidence level and population parameter: "We are 95% confident that the true difference in the population proportions of patients showing a decrease in blood pressure between the treatment and control groups..."
• 1 point: Correctly providing the context of the study: "...is between 0.021 and 0.279."

(c) Hypothesis Test Conclusion (2 points)

• 1 point: Correctly stating the conclusion: "Since 0 is not included in the interval, we can reject the null hypothesis."
• 1 point: Correctly explaining the conclusion in context: "There is evidence to suggest that the new medication is effective in reducing blood pressure."

#Answers

Multiple Choice Questions

1. (B)
2. (B)
3. (A)

That's it! You're now equipped with the knowledge to tackle confidence intervals and hypothesis testing for two proportions. Go ace that exam! 🌟