AP Statistics (AP Stats): Hypothesis Test for Difference of Two Proportions

AP Statistics – Hypothesis Test for Difference of Two Proportions

AP Statistics: Hypothesis Test for Difference of Two Proportions – Exam-Ready Study Guide

What This Is

A hypothesis test for the difference of two proportions determines whether there is convincing evidence that two population proportions differ. This test is essential on the AP exam because it applies to real-world comparisons—such as testing whether a new teaching method improves pass rates, if a drug reduces symptoms more than a placebo, or if customer satisfaction differs between two stores. You’ll use sample data to decide if observed differences are statistically significant or likely due to random chance.

Key Terms & Formulas

• Two-Proportion z-test for p? – p?:
• Test statistic: ( z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1 - \hat{p}_c) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} )
• Variables:
  • ( \hat{p}_1, \hat{p}_2 ) = sample proportions
  • ( n_1, n_2 ) = sample sizes
  • ( \hat{p}_c ) = pooled (combined) sample proportion = ( \frac{X_1 + X_2}{n_1 + n_2} ), where ( X_1, X_2 ) = number of successes in each sample

• Hypotheses:

  • ( H_0: p_1 - p_2 = 0 ) (no difference)
  • ( H_a: p_1 - p_2 \neq 0 ) (two-sided), ( p_1 - p_2 > 0 ) (one-sided), or ( p_1 - p_2 < 0 ) (one-sided)

• Pooled sample proportion (( \hat{p}_c )): Used in the denominator of the z-test because ( H_0 ) assumes ( p_1 = p_2 ).

• Conditions for inference (RIN):

• Random: Both samples are randomly selected or assigned.
• Independent: Samples are independent of each other; observations within each sample are independent (check 10% condition if sampling without replacement).

• Normal: ( n_1\hat{p}_c \geq 10 ), ( n_1(1 - \hat{p}_c) \geq 10 ), ( n_2\hat{p}_c \geq 10 ), ( n_2(1 - \hat{p}_c) \geq 10 ).

• Calculator command (TI-84):

• 2-PropZTest: Input ( x_1, n_1, x_2, n_2 ), choose ( \neq, <, ) or ( > ), and press Calculate or Draw.

• Output: z-statistic, p-value, ( \hat{p}_1, \hat{p}_2, \hat{p}_c ).

• Confidence interval for ( p_1 - p_2 ):

• Formula: ( (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{ \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} } )
• Conditions: Same as above (RIN), but do not pool ( \hat{p}_c ).

• Calculator command: 2-PropZInt.

• z* critical values (common):

• 90% CI: 1.645
• 95% CI: 1.96
• 99% CI: 2.576

Step-by-Step / Process Flow

Follow these steps for a free-response question (FRQ) on the AP exam:

1. State hypotheses in context:
2. ( H_0: p_1 - p_2 = 0 ) (no difference in [context])

3. ( H_a: p_1 - p_2 \neq 0 ) (or ( > ) or ( < )) (there is a difference in [context])

4. Check conditions (RIN):

5. Random: "Both samples were randomly selected."
6. Independent: "The samples are independent, and the 10% condition is satisfied (if applicable)."

7. Normal: Verify ( n_1\hat{p}_c \geq 10 ), etc., using the pooled proportion ( \hat{p}_c ).

8. Compute the test statistic:

9. Calculate ( \hat{p}_c ), then plug into the z-formula (or use 2-PropZTest on your calculator).

10. AP Tip: Always write the formula before plugging in numbers.

11. Find the p-value:

12. Use normalcdf(lower, upper, 0, 1) on TI-84 for two-sided tests (multiply by 2 if needed).

13. For 2-PropZTest, the p-value is automatically calculated.

14. Make a conclusion in context:

15. If p-value (e.g., 0.05): "Reject ( H_0 ). There is convincing evidence that [alternative hypothesis in context]."

16. If p-value > ?: "Fail to reject ( H_0 ). There is not convincing evidence that [alternative hypothesis in context]."

17. Link to a confidence interval (if asked):

18. If the CI for ( p_1 - p_2 ) includes 0, it supports failing to reject ( H_0 ).

Common Mistakes

• Mistake: Using ( \hat{p}_1 ) and ( \hat{p}_2 ) instead of ( \hat{p}_c ) in the denominator of the z-test.

• Correction: Always use the pooled proportion ( \hat{p}_c ) for the test statistic (but not for the confidence interval).

• Mistake: Forgetting to check the Normal condition for both samples.

• Correction: Verify ( n_1\hat{p}_c \geq 10 ), ( n_1(1 - \hat{p}_c) \geq 10 ), etc. If any are < 10, the test is invalid.

• Mistake: Mixing up the order of ( \hat{p}_1 ) and ( \hat{p}_2 ) in the hypotheses.

• Correction: Define ( p_1 ) and ( p_2 ) clearly (e.g., "( p_1 ) = proportion of Group A successes"). The order matters for one-sided tests.

• Mistake: Using a t-test instead of a z-test.

• Correction: Always use a z-test for proportions (no t-distribution involved).

• Mistake: Ignoring the 10% condition for independence.

• Correction: If sampling without replacement, check ( n \leq 0.10N ) for both samples.

AP Exam Insights

• FRQ Setup: Expect a comparison of two groups (e.g., treatment vs. control, two brands, two years). The question may ask for:
• A hypothesis test (with all steps).
• A confidence interval for ( p_1 - p_2 ).

• Interpretation of the p-value or interval in context.

• Tricky Distinction: The pooled proportion ( \hat{p}_c ) is only used in the hypothesis test, not the confidence interval.

• Calculator Pitfall: 2-PropZTest gives the p-value, but you must still state hypotheses, check conditions, and conclude in context.

• Common Trap: The AP exam may give you raw counts (e.g., "120 out of 200") instead of proportions. Convert to ( \hat{p} = \frac{X}{n} ) before plugging into formulas.

Quick Check Questions

1. Multiple Choice: A researcher tests whether a new fertilizer increases the proportion of tomato plants that survive. She randomly assigns 100 plants to the new fertilizer and 100 to the old fertilizer. In the new group, 85 survive; in the old group, 75 survive. What is the value of the pooled sample proportion ( \hat{p}_c )?
2. (A) 0.75
3. (B) 0.80
4. (C) 0.85

5. (D) 0.90 Answer: (B) 0.80. ( \hat{p}_c = \frac{85 + 75}{100 + 100} = \frac{160}{200} = 0.80 ).

6. FRQ Part: A study compares the proportion of students who pass a math test after using an online tutor (Group A) vs. a traditional tutor (Group B). In Group A (n=150), 120 pass; in Group B (n=200), 140 pass.

7. a. State the hypotheses for a test to determine if the online tutor is more effective.
8. b. Are the conditions for inference met? Justify your answer. Answer:
9. a. ( H_0: p_A - p_B = 0 ), ( H_a: p_A - p_B > 0 ) (where ( p_A ) = proportion passing with online tutor).
10. b. Yes:
    • Random: Assume random assignment.
    • Independent: Samples are independent; 10% condition is met (150-10% of all students, etc.).
    • Normal: ( n_A\hat{p}_c = 150(0.74) = 111 \geq 10 ), ( n_A(1 - \hat{p}_c) = 39 \geq 10 ), etc. (where ( \hat{p}_c = \frac{120 + 140}{350} \approx 0.74 )).

Last-Minute Cram Sheet

1. Hypotheses: ( H_0: p_1 - p_2 = 0 ); ( H_a: p_1 - p_2 \neq 0 ) (or ( > ) or ( < )).
2. Pooled proportion: ( \hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2} ) (only for hypothesis tests!).
3. Test statistic: ( z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1 - \hat{p}_c) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} ).
4. Conditions: RIN (Random, Independent, Normal: ( n_1\hat{p}_c \geq 10 ), etc.).
5. Calculator: 2-PropZTest for test; 2-PropZInt for CI.
6. Confidence interval: ( (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{ \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} } ).
7. z* values: 90% = 1.645, 95% = 1.96, 99% = 2.576.
8. Never pool for CI! Use separate ( \hat{p}_1 ) and ( \hat{p}_2 ).
9. Order matters for one-sided tests! Define ( p_1 ) and ( p_2 ) clearly.
10. Always check the 10% condition if sampling without replacement.