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Chapter 9 Hypothesis Testing II Chapter Outline Introduction Hypothesis Testing with Sample Means (Large Samples) Hypothesis Testing with Sample Means (Small Samples) Hypothesis Testing with Sample Proportions (Large Samples) Basic Logic We begin with a difference between sample statistics (means or proportions). The question we test: “Is the difference between statistics large enough to conclude that the populations represented by the samples are different?” Basic Logic The H0 is that the populations are the same. There is no difference between the parameters of the two populations If the difference between the sample statistics is large enough, so that a difference of this size is unlikely, assuming that the H0 is true, we will reject the H0 and conclude there is a difference between the populations. Basic Logic The H0 is a statement of “no difference” The 0.05 level will continue to be our indicator of a significant difference We change the sample statistics to a Z score, place the Z score on the sampling distribution and use Appendix A to determine the probability of getting a difference that large if the H0 is true. The Five Step Model 1. Make assumptions and meet test requirements. 2. State the H0. 3. Select the Sampling Distribution and Determine the Critical Region. 4. Calculate the test statistic. 5. Make a Decision and Interpret Results. Example: Hypothesis Testing in the Two Sample Case Problem 9.7b. Middle class families average 8.7 email messages/wk (s = 3.1, N=125) and working class families average 5.7(s=2.9, N=104) messages. The middle class families seem to use email more but is the difference significant? Step 1 Make Assumptions and Meet Test Requirements Model: Independent Random Samples The samples must be independent of each other. LOM is Interval Ratio Number of email messages has a true 0 and equal intervals so the mean is an appropriate statistic. Sampling Distribution is normal in shape each N is 100 or more so the Central Limit Theorem applies Step 2 State the Null Hypothesis H0: μ1 = μ2 The Null asserts there is no significant difference between the populations. Step 2 State the Alternative or Research Hypothesis H1: μ1 μ2 The research hypothesis contradicts the H0 and asserts there is a significant difference between the populations. This will be a two-tailed test. Why? Step 3 Select the S. D. and Establish the C. R. Sampling Distribution = Z distribution because N1 + N2 = 100 or more Alpha (α) = 0.05 Z (critical) = ± 1.96 We could also use a t-distribution Degrees of freedom = N1+N22=125+104=229 t (critical) = 1.96 But then we use formula 9.5 and 9.6 Step 4 Compute the Test Statistic Use Formula 9.4 to compute the pooled estimate of the standard error. Use Formula 9.2 to compute the obtained Z score. Step 5 Make a Decision The obtained test statistic (Z = 7.5) falls in the Critical Region so reject the null hypothesis. The difference between the sample means is so large that we can conclude (at α = 0.05) that a difference exists between the populations represented by the samples. ( The difference between the email usage of middle class and working class families is significant. Factors in Making a Decision The size of the difference (e.g., means of 8.7 and 5.7 for problem 9.7b) The value of alpha (the higher the alpha, the more likely we are to reject the H0 Factors in Making a Decision The use of one- vs. two-tailed tests (we are more likely to reject with a one-tailed test) The size of the sample (N). The larger the sample the more likely we are to reject the H0. Significance Vs. Importance We test for significance whenever we generalize from a sample to a population Significance is not the same thing as importance. Differences that are otherwise trivial or uninteresting may be significant. Significance Vs Importance A sample outcome could be: significant and important significant but unimportant not significant but important not significant and unimportant Test of hypothesis between two sample proportions Appropriate only with large samples (N1 +N2 = 100 or more) The logic is basically the same, but the formulae get more and more complicated See formulas 9.7, 9.8, 9.9,. 9.10 on p. 226. Again, the basic idea to establish the standard error (the standard deviation of the sampling distribution) and then do the five-step testing procedure.