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EC1011: Data Analysis II Topic 4: Hypothesis Testing Part IV: Two Populations Dr Mariya Melnychuk, [email protected]
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Outline Hypothesis testing for the difference between two population means: Two means, matched pairs Independent samples, population variances known Independent samples, population variances unknown but equal Independent samples, population variances unknown but ??? equal Hypothesis testing for the difference between two population proportions (large samples) 2
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Introduction Example 1: Consider a teacher interesting in knowing whether case studies increase students’ test scores Let’s assume she teaches two classes She can design an experiment: she assigns cases in one class and not in the other After the exam, she collect data on the mean score of the class that uses case studies and on the mean score of the other class She wants to check whether there is enough evidence that case studies help students’ scores How can she proceed? 3
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Introduction (cont.) Let 𝜇 1 denote the mean score for the population of students that use case studies Let 𝜇 2 denote the mean score for the population of students that do ??? use case studies How would you set up the null and the alternative hypotheses? Remember: to provide strong evidence that case studies increase learning we begin by assuming that case studies do not help students’ score 4
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How would you set up 𝐻 0 and 𝐻 1 ? 1. 𝐻 0 : 𝜇 1 − 𝜇 2 < 0 versus 𝐻 1 : 𝜇 1 − 𝜇 2 ≥ 0 2. 𝐻 0 : 𝜇 1 − 𝜇 2 ≥ 0 versus 𝐻 1 : 𝜇 1 − 𝜇 2 < 0 3. 𝑯 ? : 𝝁 ? − 𝝁 ? ≤ ? versus 𝑯 ? : 𝝁 ? − 𝝁 ? > ? 4. 𝐻 0 : 𝜇 1 − 𝜇 2 = 0 versus 𝐻 1 : 𝜇 1 − 𝜇 2 ≠ 0 5
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Introduction (cont.) Example 2: A reporter wants to test whether a tax reform appeals equally to men and women. Let 𝑃 ? : proportion of men favoring the proposal Let 𝑃 𝑓 : proportion of women favoring the proposal How would you set up the null and the alternative hypotheses? 1. 𝐻 0 : 𝑃 ? − 𝑃 𝑓 ≥ 0 versus 𝐻 1 : 𝑃 ? − 𝑃 𝑓 < 0 2. 𝐻 0 : 𝑃 ? − 𝑃 𝑓 ≤ 0 versus 𝐻 1 : 𝑃 ? − 𝑃 𝑓 > 0 3. 𝐻 0 : 𝑃 ? − 𝑃 𝑓 ≠ 0 versus 𝐻 1 : 𝑃 ? − 𝑃 𝑓 = 0 4. 𝑯 ? : 𝑷 ? − 𝑷 ? = ? versus 𝑯 ? : 𝑷 ? − 𝑷 ? ≠ ? 6
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Difference of Means Often we divide our data up into groups and present group means: Exam performance: male versus female Fuel efficiency: petrol versus diesel cars Graduate earnings: economics versus engineers Question: Are there significant differences between the means for each group? Conduct a hypothesis test on the difference between means 7
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Two Sample Tests Two Sample Tests Population Means, Independent Samples Population Means, Dependent Samples Group 1 vs.
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