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Week 9 Chapter 9 - Hypothesis Testing II: The Two-Sample Case

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Chapter 9 Hypothesis Testing II: The Two-Sample Case

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In This Presentation The basic logic of the two-sample case Test of significance for sample means (large and small samples) Tests of significance for sample proportions (large samples) The difference between “statistical significance” and “importance”

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In The Last Chapter Population = Penn State University Group 1 = All Education Majors RS of 100 Education Majors

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In This Chapter Population = Pennsylvania Group 1 = All Males in Population Group 2 = All Females in Population RS of 100 Males RS of 100 Females

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Basic Logic We begin with a difference between sample statistics (means or proportions) The question we test: “Is the difference between the samples large enough to allow us to conclude (with a known probability of error) that the populations represented by the samples are different?” (p. 212)

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Basic Logic The H o 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, or, if a difference of this size is unlikely, assuming that the H o is true, we will reject the H o and conclude there is a difference between the populations

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Basic Logic The H o is a statement of “no difference” The 0.05 level will continue to be our indicator of a significant difference

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The Five Step Model 1. Make assumptions and meet test requirements 2. State the null hypothesis, H o 3. Select the sampling distribution and establish the critical region 4. Calculate the test statistic 5. Make a decision and interpret the results

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The Five Step Model Changes from the one-sample case: Step 1: In addition to samples selected according to EPSEM principles, samples must be selected independently of each other: independent random sampling Step 2: Null hypothesis statement will state that the two populations are not different Step 3: Sampling distribution refers to difference between the sample statistics

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Scenario #1: Dependent variable is measured at the interval/ratio level with large sample sizes and unknown population standard deviations

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Hypothesis Test for Means Two-Sample Test of Means (Large Samples) Do men and women significantly differ on their support of gun control? For men (sample 1): Mean = 6.2 Standard deviation = 1.3 Sample size = 324 For women (sample 2): Mean = 6.5 Standard deviation = 1.4 Sample size = 317

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Step 1: Make Assumptions and Meet Test Requirements Independent random sampling The samples must be independent of each other Level of measurement is interval-ratio Support of gun control is assessed with an interval-ratio level scale, so the mean is an appropriate statistic Sampling distribution is normal in shape Total N ≥ 100 (N 1 + N 2 = 324 + 317 = 641) so the Central Limit Theorem applies and we can assume a normal shape

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Step 2: State the Null Hypothesis H o : μ 1 = μ 2 The null hypothesis asserts there is no difference between the populations H 1 : μ 1 μ 2 The research hypothesis contradicts the H o and asserts there is a difference between the populations

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Step 3: Select the Sampling Distribution and Establish the Critical Region Sampling Distribution = Z distribution Alpha (α) = 0.05 (two-tailed) Z (critical) = ±1.96

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Step 4: Compute the Test Statistic 1. Calculate the pooled estimate of the standard error 2. Calculate the obtained Z score

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Step 5: Make a Decision The obtained test statistic (Z = -2.80) falls in the critical region so reject the null hypothesis The difference between the sample means is so large that we can conclude (at alpha = 0.05) that a difference exists between the populations represented by the samples The difference between men’s and women’s support of gun control is significant

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Factors in Making a Decision The size of the difference (e.g., means of 6.2 and 6.5) The value of alpha – the higher the alpha, the more likely we are to reject the H o 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 H o

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Scenario #2: Dependent variable is measured at the interval/ratio level with small sample sizes and unknown population standard deviations

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Scenario #2 For small samples, s is too biased an estimator of σ so do not use standard normal distribution Use Student’s t distribution

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Hypothesis Test for Means Two-Sample Test of Means (Small Samples) Do families that reside in the center-city have more children than families that reside in the suburbs? For suburbs (sample 1): Mean = 2.37 Standard deviation = 0.63 Sample size = 42 For center-city (sample 2): Mean = 2.78 Standard deviation = 0.95 Sample size = 37

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Step 1: Make Assumptions and Meet Test Requirements Independent random sampling The samples must be independent of each other Level of measurement is interval-ratio Number of children can be treated as interval-ratio Population variances are equal As long as the two samples are approximately the same size, we can make this assumption Sampling distribution is normal in shape Because we have two small samples (N < 100), we have to add the previous assumption in order to meet this assumption

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Step 2: State the Null Hypothesis H o : μ 1 = μ 2 The null hypothesis asserts there is no difference between the populations H 1 : μ 1 < μ 2 The research hypothesis contradicts the H o and asserts there is a difference between the populations

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Step 3: Select the Sampling Distribution and Establish the Critical Region Sampling Distribution = t distribution Alpha (α) = 0.05 (one-tailed) Degrees of freedom = N 1 + N 2 -2 = 77 t (critical) = -1.671

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Step 4: Compute the Test Statistic 1. Calculate the pooled estimate of the standard error 2. Calculate the obtained Z score

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Step 4: Compute the Test Statistic

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Step 5: Make a Decision The obtained test statistic (t = -2.16) falls in the critical region so reject the null hypothesis The difference between the sample means is so large that we can conclude (at alpha = 0.05) that a difference exists between the populations represented by the samples The difference between the number of children in center-city families and the suburban families is significant

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Scenario #3: Dependent variable is measured at the nominal level with large sample sizes and unknown population standard deviation

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Hypothesis Test for Proportions Two-Sample Test of Proportions (Large Samples) Do Black and White senior citizens differ in their number of memberships in clubs and organizations? (Using the proportion of each group classified as having a “high” level of membership) For Black senior citizens (sample 1): Proportion = 0.34 Sample size = 83 For White senior citizens (sample 2): Proportion = 0.25 Sample size = 103

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Step 1: Make Assumptions and Meet Test Requirements Independent random sampling The samples must be independent of each other Level of measurement is nominal We have measured the proportion of each group classified as having a “high” level of membership Sampling distribution is normal in shape Total N ≥ 100 (N 1 + N 2 = 83 + 103 = 186) so the Central Limit Theorem applies and we can assume a normal shape

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Step 2: State the Null Hypothesis H o : P u1 = P u2 The null hypothesis asserts there is no difference between the populations H 1 : P u1 ≠ P u2 The research hypothesis contradicts the H o and asserts there is a difference between the populations

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Step 3: Select the Sampling Distribution and Establish the Critical Region Sampling Distribution = Z distribution Alpha (α) = 0.05 (two-tailed) Z (critical) = ±1.96

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Step 4: Compute the Test Statistic 1. Calculate the pooled estimate of the standard error

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Step 4: Compute the Test Statistic 2.Calculate the obtained Z score

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Step 5: Make a Decision The obtained test statistic (Z = +1.29) does not fall in the critical region so fail to reject the null hypothesis The difference between the sample means is small enough that we can conclude (at alpha = 0.05) that no difference exists between the populations represented by the samples The difference between the memberships of Black and White senior citizens is not significant

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Limitations of Hypothesis Testing Significance Versus Importance As long as we work with random samples, we must conduct a test of significance Significance is not the same thing as importance Differences that are otherwise trivial or uninteresting may be significant

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Conclusion Scenario #1: Dependent variable is measured at the interval/ratio level with large sample sizes and unknown population standard deviations Use standard Z distribution formula Scenario #2: Dependent variable is measured at the interval/ratio level with small sample sizes and unknown population standard deviations Use standard T distribution formula Scenario #3: Dependent variable is measured at the nominal level with large sample sizes and known population standard deviations Use slightly modified Z distribution formula Scenario #4: Dependent variable is measured at the nominal level with small sample sizes This is not covered in your text or in class

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USING SPSS On the top menu, click on “Analyze” Select “Compare Means” Select “Independent Samples T Test OR Paired Samples T Test”

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Test on two means (independent samples) Analyze / Compare means / Independent samples t-test Specify which groups you are comparing

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Output The null hypothesis is rejected (as the p-value is smaller than 0.05)

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