AP STATISTICSLESSON 12 – 2 ( DAY 2 )

SIGNIFICANCE TESTS FOR p1 – p2

ESSENTIAL QUESTION: How are significance tests created for two sample population proportions?

Objectives: To learn the procedures for finding

significance tests. To be able to pool sample

proportions.

Significance Tests for p1 – p2

The null hypothesis says that there is no difference between the two populations:

Ho: p1 = p2 or Ho: p1 – p2 = 0

The alternative hypothesis says what kind of difference we expect.

Example 12.12 Page 707Cholesterol and Heart Attacks

To do a test, standardize p1 – p2 to get a z statistic.

If H is true, all the observations in both samples really come from a single population of men of whom a single unknown proportion p will have a heart attack in five-year period.

So instead of estimating p1 and p2 separately, we pool the two samples and use the overall sample proportion to estimate the single population parameter p.

Call this the pooled sample proportion.

^^

Pooled Sample Proportion

p = count of success in both samples combined = X1 + X2

count of observations in both samples combined n1 + n2

^

Significance Test for Comparing Two Proportions

To test the hypothesis:

H0 : p1 = p2

first find the pooled proportion p of successes in both samples combine. Then compute the z statistic

z = p1 – p2

√ p( 1 – p ) ( 1/n1 + 1/ n2 )

^^^

^

P-values In terms of a variable Z having the standard normal

distribution, the P-value for a test Ho against

Ha: p1 > p2 is P(Z ≥ z )

Ha: p1 < p2 is P(Z ≤ z )

Ha: p1 ≠ p2 is 2P(Z ≥ lzl )

Conditions: Use these tests when the populations are at least 10 times as large as the samples and n 1p, n 1(1 – p), n 2p and n 2( 1 – p ) are all 5 or more. ^^

^^

Example 12.13 Page 709Cholesterol and Heart

Attacks (continued…)

p = count of heart attacks in both samples combined

count of observations in both samples combined

^

Since P < 0.01, the results are statistically significant at the a = 0.01 level.

There is strong evidence that the Gemfibrozil reduced the rate of heart attacks.

Example 12.14 Page 710Don’t Drink the Water

The P-value, 0.0344 tells us that it is unlikely that we would obtain a difference in sample proportions as large as we did if the null hypothesis is true.

Judges have generally adopted a 5% significance level as their standard for convincing evidence.

The P-value for the one-sided test