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Understanding Significance vs. Importance in Hypothesis Testing

Learn the difference between statistical significance and importance in hypothesis testing, with examples and the basic logic for two-sample tests of significance for sample means. Understand the Five Step Model for hypothesis testing along with practical examples.

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Understanding Significance vs. Importance in Hypothesis Testing

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

  2. 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?”

  3. 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, or, if a difference of this size is unlikely, assuming that the H0 is true, we will rejectthe H0 and conclude there is a difference between the populations.

  4. 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.

  5. The Five Step Model • Make assumptions and meet test requirements. • State the H0. • Select the Sampling Distribution and Determine the Critical Region. • Calculate the test statistic. • Make a Decision and Interpret Results.

  6. Example: Hypothesis Testing in the Two Sample Case • Problem 9.5b (p. 243 in Healey) • Middle class families average 8.7 email messages and working class families average 5.7 messages. • The middle class families seem to use email more but is the difference significant?

  7. 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 • N = 144 cases so the Central Limit Theorem applies and we can assume a normal shape.

  8. Step 2 State the Hypotheses • H0: μ1 = μ2 • The Null hypothesis asserts there is no significant difference between the populations. • H1: μ1 μ2 • The alternative, research hypothesis contradicts the H0 and asserts there is a significant difference between the populations.

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

  10. 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.

  11. 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 email usage of middle class and working class families is significant.

  12. 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 • 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.

  13. 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.

  14. 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.

  15. 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

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