Chapter 9 Hypothesis Testing II: two samples Test of significance for sample means (large samples) The difference between “statistical significance” and “importance”.

Basic Logic of the two sample case  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 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.

Basic Logic  The H 0 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 H 0 is true.

The Five Step Model 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.

Example: Hypothesis Testing in the Two Sample Case  Problem 9.5b (p. 243 in Healey) Middle class families average 8.7 messages and working class families average 5.7 messages. The middle class families seem to use 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 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 2 State the Hypotheses  H 0 : μ1 = μ2 The Null hypothesis asserts there is no significant difference between the populations.  H 1 : μ1  μ2 The alternative, research hypothesis contradicts the H 0 and asserts there is a significant difference between the populations.

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

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 = 20.00) 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 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 H 0  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 0.

Significance Vs. 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.

Significance Vs. Importance  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  Significance and importance are different things.  In general, when working with random samples, 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