#AP Statistics: Hypothesis Testing for Two Proportions - The Night Before 🏀

Hey, future AP Stats superstar! Let's get you feeling confident and ready to crush this exam. We're going to break down hypothesis testing for two proportions, focusing on what you really need to know. Let's do this! 💪

#Hypothesis Testing for Two Proportions

This section focuses on comparing two population proportions to see if there is a statistically significant difference between them. It's a big topic, so let's make sure you've got it down. This is a high-value topic, so it's worth your time! 💯

#Calculating Values

To perform a hypothesis test for the difference in two population proportions, we need to calculate two key values: the z-score and the p-value. These are crucial for making a decision about our null hypothesis.

#Z-Score

The z-score is our test statistic. It tells us how many standard deviations our sample difference is from the hypothesized mean (usually zero). The formula looks intimidating, but don't worry, your calculator does the heavy lifting! 🏋️

latex
z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}

• $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions.

• $p_1 - p_2$ is the hypothesized difference (usually 0).

• $n_1$ and $n_2$ are the sample sizes.

#P-Value

The p-value is the probability of observing a sample difference as extreme as (or more extreme than) the one we got, assuming the null hypothesis is true. It's like asking, "How likely is this result if there's really no difference?" 🧐

#Concluding the Test

Once we have the z-score and p-value, we decide whether to reject the null hypothesis (H₀). Here’s how:

#Using P-Value

• Compare the p-value to the significance level (α), usually 0.05. * If the p-value is LOW, reject the H₀. (Low p-value = surprising result if H₀ is true).

*   **p-value < α:** Reject H₀. There's strong evidence against the null hypothesis.

*   **p-value ≥ α:** Fail to reject H₀. There isn't enough evidence to reject the null hypothesis.

#Using Z-Score

• Use the empirical rule (68-95-99.7 rule) to assess the z-score's extremity.
• High z-score = rare event if H₀ is true = reject H₀.

*   **|z| > 2 or 3:** Reject H₀. The result is very unusual if the null is true.

*   **|z| < 2 or 3:** Fail to reject H₀. The result isn't unusual if the null is true.

#Continued Example: MJ vs. Lebron 🏀

Let's revisit the classic debate: Is Michael Jordan a better shooter than Lebron James? We'll use our hypothesis testing skills to see if our data supports this claim.

• MJ: 836/1623 shots made
• Lebron: 622/1493 shots made

#Calculations

We'll use the calculator's 2-PropZTest function to get our z-score and p-value. Here's what it looks like on the calculator:

#Conclusion

#P-Value

Our calculator gives us a p-value of approximately 0. This is very low! We can conclude:

"Since our p-value is less than 0.05 (~0 < 0.05), we reject our H₀. We have convincing evidence that the true population proportion of shots made by MJ is higher than the true population proportion of shots made by Lebron."

#Z-Score

Our z-score is about 5.5, which is way beyond 2 or 3. This also leads us to reject the null hypothesis. 💡

#Final Exam Focus

• Hypothesis Testing Steps: State hypotheses, check conditions, calculate test statistic (z-score), find p-value, make a conclusion in context.

• Conditions for Inference: Random samples, independence, large enough sample sizes (np ≥ 10 and n(1-p) ≥ 10 for each group).

• Interpreting P-values: Understand what a p-value represents and how it relates to the null hypothesis.

• Type I and Type II Errors: Be able to define them and understand their consequences.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. Move on and come back if you have time.
• Calculator Skills: Know how to use your calculator for hypothesis tests and confidence intervals.
• Read Carefully: Pay close attention to wording in the problem. What are they really asking?
• Show Your Work: Even if you use your calculator, show the steps you're taking.
• Stay Calm: You've got this! Take deep breaths and trust your preparation.

#Practice Questions

Practice Question

Multiple Choice Questions

1. A researcher is testing the hypothesis that the proportion of left-handed people is different in two different countries. They collect random samples from each country. Which of the following is the most appropriate test? (A) One-sample z-test for a proportion (B) Two-sample z-test for proportions (C) One-sample t-test for a mean (D) Two-sample t-test for means (E) Chi-square test for independence

2. A two-sample z-test for proportions is conducted with a null hypothesis that the two proportions are equal. The calculated test statistic is z = 2.15. Which of the following is the correct conclusion at α = 0.05? (A) Reject the null hypothesis; there is significant evidence that the two proportions are different. (B) Fail to reject the null hypothesis; there is not significant evidence that the two proportions are different. (C) Reject the null hypothesis; there is not significant evidence that the two proportions are different. (D) Fail to reject the null hypothesis; there is significant evidence that the two proportions are different. (E) There is not enough information to make a conclusion.

3. In a study comparing the effectiveness of two different medications, researchers found a p-value of 0.08. If the significance level is set at 0.05, what is the correct conclusion? (A) Reject the null hypothesis. (B) Fail to reject the null hypothesis. (C) There is not enough information to make a conclusion. (D) The study proves the null hypothesis is false. (E) The study proves the null hypothesis is true.

Free Response Question

A large school district is trying to decide whether to implement a new math curriculum. They randomly select 200 students from high school A and 250 students from high school B. The students take a standardized math test. 120 of the students from school A and 160 of the students from school B passed the test. At a significance level of 0.05, is there a significant difference in the proportion of students who passed the test between the two schools?

Scoring Guide:

Part 1: Hypotheses (1 point)

• Correctly states the null and alternative hypotheses. * H₀: p₁ = p₂ (or p₁ - p₂ = 0) i.e. there is no difference in the proportions of students who passed the test between the two schools. * Hₐ: p₁ ≠ p₂ (or p₁ - p₂ ≠ 0) i.e. there is a difference in the proportions of students who passed the test between the two schools.

Part 2: Conditions (1 point)

• Checks the conditions for a two-sample z-test for proportions. * Random samples: The problem states that students were randomly selected from each school. * Independence: The samples are independent because the students from each school are not related. * Large enough sample sizes: n₁p₁ = 200 * (120/200) = 120, n₁(1-p₁) = 200 * (80/200) = 80, n₂p₂ = 250 * (160/250) = 160, n₂(1-p₂) = 250 * (90/250) = 90. All are greater than 10. Part 3: Calculations (2 points)

• Correctly calculates the test statistic (z-score) and p-value. * z = (0.6 - 0.64) / sqrt((0.6 * 0.4)/200 + (0.64 * 0.36)/250) = -0.926 * p-value = 0.354

Part 4: Conclusion (1 point)

• Makes a correct conclusion in the context of the problem. * Since the p-value (0.354) is greater than the significance level (0.05), we fail to reject the null hypothesis. There is not sufficient evidence to conclude that there is a difference in the proportions of students who passed the test between the two schools.

You've got this! Go get that 5! 🎉