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AP STATISTICS LESSON 12 – 2 ( DAY 2 ) SIGNIFICANCE TESTS FOR p 1 – p 2

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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.

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Significance Tests for p 1 – p 2 The null hypothesis says that there is no difference between the two populations: H o : p 1 = p 2 or H o : p 1 – p 2 = 0 The alternative hypothesis says what kind of difference we expect.

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Example Page 707 Cholesterol and Heart Attacks To do a test, standardize p 1 – p 2 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 p 1 and p 2 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. ^^

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Pooled Sample Proportion p = count of success in both samples combined = X 1 + X 2 count of observations in both samples combined n 1 + n 2 ^

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Significance Test for Comparing Two Proportions To test the hypothesis: H 0 : p 1 = p 2 first find the pooled proportion p of successes in both samples combine. Then compute the z statistic z = p 1 – p 2 √ p( 1 – p ) ( 1/n 1 + 1/ n 2 ) ^ ^^ ^

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P-values In terms of a variable Z having the standard normal distribution, the P-value for a test H o against H a : p 1 > p 2 is P(Z ≥ z ) H a : p 1 < p 2 is P(Z ≤ z ) H a : p 1 ≠ p 2 is 2P(Z ≥ lzl ) Conditions: Use these tests when the populations are at least 10 times as large as the samples and n 1 p, n 1 (1 – p), n 2 p and n 2 ( 1 – p ) are all 5 or more. ^ ^ ^ ^ ^

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Example Page 709 Cholesterol 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.

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Example Page 710 Don’t Drink the Water The P-value, 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

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