1. Copyright © 2012 by Nelson Education Limited. Chapter 8 Hypothesis Testing II: The Two-Sample Case 8-1

2. Copyright © 2012 by Nelson Education Limited. The basic logic of the two sample case. Hypothesis Testing with Sample Means (Large Samples), Sample Means (Small Samples), and Sample Proportions (Large Samples) The difference between “statistical significance” and “importance” In this presentation you will learn about: 8-2

3. Copyright © 2012 by Nelson Education Limited. 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?” Hypothesis Test for Two Samples: Basic Logic 8-3

4. Copyright © 2012 by Nelson Education Limited. The null hypothesis, H 0, 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 0 is true, we will reject the H 0 and conclude there is a difference between the populations. Hypothesis Test for Two Samples: Basic Logic (continued) 8-4

5. Copyright © 2012 by Nelson Education Limited. The H 0 is a statement of “no difference.” The 0.05 level will continue to be our indicator of a significant difference. Sample means or proportions with large samples (combined n’s >100) 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 H 0 is true. Hypothesis Test for Two Samples: Basic Logic (continued) 8-5

6. Copyright © 2012 by Nelson Education Limited. Sample means with small samples we: –change the sample statistic to a t score, place the t score on the sampling distribution, and use Appendix B to determine the probability of getting a difference that large if the H 0 is true. Hypothesis Test for Two Samples: Basic Logic (continued) 8-6

7. Copyright © 2012 by Nelson Education Limited. 1.Make assumptions and meet test requirements. 2.State the H 0. 3.Select the Sampling Distribution and Determine the Critical Region. 4.Calculate the test statistic. 5.Make a Decision and Interpret Results. Testing Hypotheses: The Five Step Model 8-7

8. Copyright © 2012 by Nelson Education Limited. Step 1: in addition to samples selected according to EPSEM principles, samples must be selected independently: Independent random sampling. Step 2: null hypothesis statement will say the two populations are not different. Step 3: sampling distribution refers to difference between the sample statistics. Changes from One- to Two- Sample Case 8-8

9. Copyright © 2012 by Nelson Education Limited. Steps 4 and 5 are same as before: Step 4: In computing the test statistic, Z(obtained) or t(obtained), the form of the formula remains the same. Step 5: same as before: If the test statistic, Z(obtained) or t(obtained), falls into the critical region, as marked by Z(critical) or t(critical), reject the H 0. Changes from One- to Two- Sample Case (continued) 8-9

10. Copyright © 2012 by Nelson Education Limited. Do middle- and working-class families differ in their use of email to maintain family contact? o The data below report the average number of email messages per month with close kin for random samples of middle and working class families: The middle class seem to use email more than the working class, but is the difference significant? Hypothesis Test for Two- Sample Means (Large Samples) 8-10

11. Copyright © 2012 by Nelson Education Limited. Model: –Independent Random Samples The samples must be independent of each other. –Level of Measurement 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 N = 144 cases so the Central Limit Theorem applies and we can assume a normal shape. Step 1: Make Assumptions and Meet Test Requirements 8-11

12. Copyright © 2012 by Nelson Education Limited. No direction for the difference has been predicted, so a two-tailed test is called for, as reflected in the research hypothesis: –H 0 : μ 1 = μ 2 The Null asserts there is no significant difference between the populations. –H 1 : μ 1  μ 2 The research hypothesis contradicts the H 0 and asserts there is a significant difference between the populations. Step 2: State the Null Hypothesis 8-12

13. Copyright © 2012 by Nelson Education Limited. Sampling Distribution = Z distribution Alpha (α) = 0.05 Z (critical) = ± 1.96 Step 3: Select Sampling Distribution and Establish the Critical Region 8-13

14. Copyright © 2012 by Nelson Education Limited. Use Formula 8.2 to compute the obtained Z score: where is the standard deviation of the sampling distribution. Step 4: Compute the Test Statistic 8-14

15. Copyright © 2012 by Nelson Education Limited. When the population standard deviations are known, we use Formula 8.3 to calculate : and Formula 8.4 when they are unknown: In Formula 8.4 we use s as an estimator of σ, suitably corrected for the bias (n is replaced by n-1 to correct for the fact that s is a biased estimator of σ). Sample size must be large (combined n’s >100). Step 4: Compute the Test Statistic (continued) 8-15

16. Copyright © 2012 by Nelson Education Limited. where Z (obtained) == = Step 4: Compute the Test Statistic (continued) = = == = 8-16

17. Copyright © 2012 by Nelson Education Limited. The obtained test statistic (Z = 19.7) 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 email usage of middle- and working-class families is significant. Step 5: Make Decision and Interpret Results 8-17

18. Copyright © 2012 by Nelson Education Limited. For small samples (combined n’s<100), s is too unreliable an estimator of σ so do not use standard normal distribution. Instead we use Student’s t distribution. The formula for computing the test statistic, t(obtained), is: where is defined as: Hypothesis Test for Two-Sample Means: Student’s t distribution (Small Samples) 8-18

19. Copyright © 2012 by Nelson Education Limited. The logic of the five-step model for hypothesis testing is followed, using the t table, Appendix B, where the degrees of freedom (df) = n 1 + n 2 – 2. Hypothesis Test for Two-Sample Means: Student’s t distribution (continued) 8-19

20. Copyright © 2012 by Nelson Education Limited. We can also use Z (obtained) to test sample proportions, as long as sample size is large (combined n’s >100): The logic of the five-step model for hypothesis testing is followed. *Small-sample tests of hypothesis for proportions are not considered in this text. Test of Hypothesis for Two- Sample Proportions (Large Samples)* 8-20

21. Copyright © 2012 by Nelson Education Limited. The probability of rejecting the null hypothesis is a function of four independent factors: 1.The size of the difference (e.g., means of 8.7 and 5.7 for the example above). 2.The value of alpha (the higher the alpha, the more likely we are to reject the H 0 ). 3.The use of one- vs. two-tailed tests (we are more likely to reject with a one-tailed test). 4.The size of the sample (n) (the larger the sample the more likely we are to reject the H 0 ). Significance vs. Importance 8-21

22. Copyright © 2012 by Nelson Education Limited. As long as we work with random samples, we must conduct a test of significance. However, significance is not the same thing as importance. Significance vs. Importance (continued) 8-22

23. Copyright © 2012 by Nelson Education Limited. –Differences that are otherwise trivial or uninteresting may be significant, which is a major limitation of hypothesis testing. ◦ When working with large samples, even small differences may be significant. ◦ The value of the test statistic (step 4) is an inverse function of n. ◦ The larger the n, the greater the value of the test statistic, the more likely it will fall in the Critical Region and be declared significant. Significance vs. Importance (continued) 8-23

24. Copyright © 2012 by Nelson Education Limited. In conclusion, significance is a necessary but not sufficient condition for importance. A sample outcome could be: –significant and important –significant but unimportant –not significant but important –not significant and unimportant Significance vs. Importance (continued) 8-24