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# lecture8 - ECO201 Lecture8 HypothesisTesting ZandtTests...

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ECO 201 Lecture 8 Hypothesis Testing:   and  t   Tests

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Topics Tests for the Difference between Two  Proportions Testing for the Difference Between the Means  of Two Independent Groups Paired  t   Test
Two-Sample Tests Two-Sample Tests Population Means, Independent Samples Means, Related Samples Population 1 vs. independent Population 2 Same population before vs. after treatment Examples: Population Proportions Proportion 1 vs. Proportion 2

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Hypothesis Tests for Two Population Proportions Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π 1 π 2 The _____________ for the difference is Population proportions Assumptions: n 1 π 1 5 , n 1 (1- π 1 ) 5 n 2 π 2 5 , n 2 (1- π 2 ) 5 2 1 p p -
Confidence Interval for Two Population Proportions Population proportions ( 29 2 2 2 1 1 1 2 1 n ) p (1 p n ) p (1 p Z p p - + - ± - The confidence interval for π 1 π 2 is:

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Hypothesis Tests for Two Population Proportions Population proportions tail test: H 0 : π 1 π 2 H 1 : π 1 < π 2 i.e., H 0 : π 1 π 2 0 H 1 : π 1 π 2 < 0 tail test: H 0 : π 1 π 2 H 1 : π 1 > π 2 i.e., H 0 : π 1 π 2 ≤ 0 H 1 : π 1 π 2 > 0 Two-tail test: H 0 : π 1 = π 2 H 1 : π 1 π 2 i.e., H 0 : π 1 π 2 = 0 H 1 : π 1 π 2 ≠ 0
Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : π 1 π 2 0 H 1 : π 1 π 2 < 0 Upper-tail test: H 0 : π 1 π 2 ≤ 0 H 1 : π 1 π 2 > 0 Two-tail test: H 0 : π 1 π 2 = 0 H 1 : π 1 π 2 ≠ 0 α α /2 α /2 α -z α -z α /2 z α z α /2 Reject H 0 if Z < -Z α Reject H 0 if Z > Z α Reject H 0 if Z < -Z α/2 or Z > Z α/2 (continued)

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Example:  Z Test for Difference in Two population  Proportions    Is there a significant difference between the  proportion of men and the proportion of women who  will vote Yes on Proposition A? In a random sample, 36 of 72 men and 31 of 50  women indicated they would vote Yes Test at the .05 level of significance
The hypothesis test is: H 0 : π 1 π 2 = (the two proportions are equal) H 1 : π 1 π 2 (there is a significant difference between proportions) The sample proportions are: Men:   p 1  = 36/72 = .50 Women:   p 2  = 31/50 = .62 .549 122 67 50 72 31 36 n n X X p 2 1 2 1 = = + + = + + = The pooled estimate for the overall proportion is: Example:  Z Test for Difference in Two population  Proportions (continued)

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10 Z Test for the Difference Between  Two Proportions Add-Ins (tab)|Menu Commands|PHStat  |  Two-Sample Tests  |  Z Test for Differences  in Two Proportions …   Double Click Here  for Excel Output
Excel output: Z Test for Differences in Two Proportions Hypothesized Difference 0 Level of Significance 0.05 Group 1 Number of Successes 36 Sample Size 72 Group 2 Number of Successes 31 Sample Size 50 Group 1 Proportion 0.5 Group 2 Proportion 0.62 Difference in Two Proportions -0.12 Average Proportion 0.549180328 Z Test Statistic -1.310067478

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lecture8 - ECO201 Lecture8 HypothesisTesting ZandtTests...

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