Review of Chapter 11 Comparison of Two Populations Lesson 11 – R Review of Chapter 11 Comparison of Two Populations

Similarities in Hypothesis Test Processes Parameter Two Means (Independent) (Dependent) Two Proportions Two Variances Std Devs H0: μ1 = μ2 p1 = p2 σ12 = σ22 σ1 = σ2 (2-tailed) H1: μ1 ≠ μ2 p1 ≠ p2 σ12 ≠ σ22 σ1 ≠ σ2 (L-tailed) H1: μ1 < μ2 p1 < p2 σ12 < σ22 σ1 < σ2 (R-tailed) H1: μ1 > μ2 p1 > p2 σ12 > σ22 σ1 > σ2 Test statistic Difference Ratio Critical value Student t Normal F

Putting it All Together Provided each sample size ≥ 30 or differences come from population that is normally distributed, use t-distribution with n-1 degrees of freedom and Provided np(1-p) ≥ 10 and n ≤ 0.05N for each sample, use normal distribution with (x1 – x2) – (μ1 – μ2 ) t0 = ------------------------------- s12 s22 ----- + ----- n1 n2 p1 – p2 z0 = --------------------------------- p (1- p) where x1 + x2 p = ------------ n1 + n2 1 1 --- + --- n1 n2 proportion, p Independent What parameters are addressed in the claim? Dependent or Independent Sampling μ Dependent s or s2 Provided each sample size ≥ 30 or differences come from population that is normally distributed, use t-distribution with n-1 degrees of freedom and Provided data are normally distributed, use F-distribution with s12 F0 = ------------ s22 d – μd t0 = ------------ sd / n

Multiple Choice Evaluations Which of the following is a characteristic of dependent (or matched-pairs) samples? The observations from the sample are matched to the hypothesis being tested The observations from sample 1 and sample 2 are paired with each other The observations from sample 1 are independent of the observations from sample 2 The mean of sample 2 depends on the mean of sample 1 Section 1

Multiple Choice Evaluations When we have dependent (or matched-pairs) data, then we should Test the equality of their squares using a chi-square distribution Test the equality of their means using proportions Test the equality of their variances using a normal distribution Test their differences using a student t-distribution Section 1

Multiple Choice Evaluations A researcher collected data from two sets of patients, both chosen at random from a large population of patients. If there is no interaction between the two groups, then this is an example of Independent samples Dependent samples Stratified sampling Descriptive statistics Section 2

Multiple Choice Evaluations The standard deviation for the difference of two means from independent samples involves The standard deviation of each sample The mean of each sample The difference between the two means All of the above Section 2

Multiple Choice Evaluations To compare two population proportions, we Multiply the two proportions and take the square root Subtract one proportion from the other and divide by the appropriate standard deviation Subtract one proportion from the other and use the chi-square distribution Divide one proportion by the other and use the normal distribution Section 3

Multiple Choice Evaluations If a researcher requires 100 subjects in sample 1 and 100 subjects in sample 2 to achieve a particular margin of error, then the number of subjects required to halve the margin of error is 200 subjects in each of sample 1 and sample 2 200 subjects in sample 1 and 100 subjects in sample 2 50 subjects in each of sample 1 and sample 2 400 subjects in each of sample 1 and sample 2 Section 3

Multiple Choice Evaluations To test for the equality of two population standard deviations, we use for the test statistic The difference between the two standard deviations The ratio of the two standard deviations The product of the two standard deviations The ratio of the squares of the two standard deviations Section 4

Multiple Choice Evaluations To test for the equality of two population standard deviations, we use The normal distribution The beta distribution The F distribution The chi-square distribution Section 4

Summary and Homework Summary Homework We can test whether sample data from two different samples supports a hypothesis claim about a population mean, proportion, or standard deviation For two population means, there are two cases Dependent (or matched-pair) samples Independent samples All of these tests follow very similar processes as hypothesis tests on one sample Homework pg 625 – 628; 1, 4, 6, 7, 10, 11, 14, 15, 21

Homework Answers 4) Dependent 6) a) 2.06 b) 0.49 10) a) Reject H0 b) [-12.44, 0.04] c) Reject H0 14) a) H0: p1 = p2 H1: p1 < p2 b) z0 = -0.62 c) zc = -1.645 d) p = 0.2611 FTR H0