Decision making for two samples Inference about two population means

From original data Z: from N(0,1) or T: from T-dist Large sample size or from Normal population Caution: the multiplier depends on the significance level

Formula for p-values From H 0 From original data Compare z to N(0,1) or t to T distribution for p-value Large sample size or from Normal population Caution: The direction of the tail depends on alternative hypothesis

Decision making for two samples ParameterDistributionStandard Error (CI)Standard Error (Test) Difference in ProportionsNormal Difference in Means Variance Known Normal Difference in Means Variance Unknown but same Pooled Difference in Means Variance Unknown but diff. Unpooled t, df = min(n 1, n 2 ) – 1 Difference in Means (Paired)t, df = n – 1

Reading speed Do you think that reading speed is related to the arrangement of the text. a) Yes b) No c) Not Sure

Reading speed Do you think that reading speed is related to the arrangement of the text. a) A b) B c) Not Sure A B

Reading speed How can we perform an experiment to test our hypothesis. We randomly select some liberal arts students to read horizontal text, and randomly select some science students to read vertical text. Then compare their reading speed. a) Valid b) Invalid c) Not Sure

Reading speed How can we perform an experiment to test our hypothesis. We randomly select people from the street and randomly partition these people into two groups. Let them read the same text but one group reads vertical arrangement the other group reads horizontal arrangement, Then compare their reading speed. a) Valid b) Invalid c) Not Sure

Reading speed We want to test the hypothesis that reading vertical text is slower. μ 1 = reading time of vertical text μ 2 = reading time of horizontal text a) H 0 : μ 1 = μ 2, H a : μ 1 ≠ μ 2 b) H 0 : μ 1 = μ 2, H a : μ 1 > μ 2 c) H 0 : μ 1 = μ 2, H a : μ 1 < μ 2 H 0 : μ 1 - μ 2 = 0, H a : μ 1 - μ 2 ≠ 0 H 0 : μ 1 - μ 2 = 0, H a : μ 1 - μ 2 > 0 H 0 : μ 1 - μ 2 = 0, H a : μ 1 - μ 2 < 0 Null Value Null value can be a number, then we are testing hypothesis that μ 1 take more seconds than μ 2

Inference about two population means - Now we obtained two independent random samples. - We record the reading time from group 1, and reading time from group 2. The diff. Test Stat.

Assume Variances Known ParameterDistributionStandard Error (CI)Standard Error (Test) Difference in ProportionsNormal Difference in Means (Variance Known) Normal Difference in Means (Variance Unknown but same Pooled) Difference in Means (Variance Unknown but diff. Unpooled ) t, df = min(n 1, n 2 ) – 1 Difference in Means (Paired)t, df = n – 1

Assume Population Variances are Unknown. Then we need to estimate the variances of two populations. Two ways: 1. Assume two populations have same variance 2. Assume two populations have different variances.

Case 1. Assume two populations have the same variance.

Case 1. Assume two populations have the same variance. Then we use one sample variance across two samples for a common population variance

ParameterDistributionStandard Error (CI)Standard Error (Test) Difference in ProportionsNormal Difference in Means Variance Known Normal Difference in Means Variance Unknown but same Pooled Difference in Means Variance Unknown but diff. Unpooled t, df = min(n 1, n 2 ) – 1 Difference in Means (Paired)t, df = n – 1 Case 1. Assume two populations have the same variance.

Case 2. Assume two populations have different variances.

Case 2. Assume two populations have different variances.

ParameterDistributionStandard Error (CI)Standard Error (Test) Difference in ProportionsNormal Difference in Means Variance Known Normal Difference in Means Variance Unknown but same Pooled Difference in Means Variance Unknown but diff. Unpooled t, df = min(n 1, n 2 ) – 1 Difference in Means (Paired)t, df = n – 1

Inference about two population means -We need to hire two large groups of people for our experiment -Large sample size comes with high cost. -Can we test our hypothesis without having two groups? YES

Decision making for two samples ParameterDistributionStandard Error (CI)Standard Error (Test) Difference in ProportionsNormal Difference in Means Variance Known Normal Difference in Means Variance Unknown but same Pooled Difference in Means Variance Unknown but diff. Unpooled t, df = min(n 1, n 2 ) – 1 Difference in Means (Paired)t, df = n – 1

Confidence interval for two population means - Now we obtained two independent random samples. - We record the reading time from group 1, and reading time from group 2. Multi.

Inference about two population means - For the same group of people -Each person read vertical and horizontal text. -Then we compare the reading time for EACH INDIVIDUAL. D 1 = difference for the first individual … D n = difference for the n th individual Although native English speakers normally read horizontal text, there are…