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MATH& 146 Lesson 32 Practical Significance 1
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1. 1. MATH& 146 Lesson 32 Practical Significance 1
2. 2. Example 1 Suppose a random sample of size 24 had a sample mean of 33.8 and a sample standard deviation of 9.8. a) Test the hypotheses Use a significance level of 0.05. b) Find the 95% confidence interval to estimate . Where is the null value of 30 on this interval? 2 0 : 30, : 30.AH H
3. 3. Example 2 Suppose a random sample of size 30 had a sample mean of 32.8 and a sample standard deviation of 10.57. a) Test the hypotheses Use a significance level of 0.01. b) Find the 99% confidence interval to estimate . Where is the null value of 40 on this interval? 3 0 : 40, : 40.AH H
4. 4. Example 3 Suppose a random sample of size 10 had a sample mean of 19.9 and a sample standard deviation of 6.85. a) Test the hypotheses Use a significance level of 0.05. b) Find the 95% confidence interval to estimate . Where is the null value of 15 on this interval? 4 0 : 15, : 15.AH H
5. 5. Confidence Intervals and Tests There is a close relationship between confidence intervals and significance tests. Specifically, if a point estimate is significantly different from the null value at the 0.05 level, then the 95% confidence interval will not contain that null value. 5
6. 6. Confidence Intervals and Tests Values inside the confidence interval are plausible values for the parameter, whereas values outside the interval can be considered implausible. 6 (------------------------------|------------------------------) Point Estimate Null values inside interval mean "fail to reject". Null values outside interval mean "reject".
7. 7. Confidence Intervals and Tests Looking at non-significant effects in terms of confidence intervals makes clear why the null hypothesis should not be accepted when it is not rejected: Every value in the confidence interval is a plausible value of the parameter. If a null value is in the interval, then it is plausible and cannot be rejected. However, there is an infinite number of other values in the interval (assuming continuous measurement), and none of them can be rejected either. 7
8. 8. Example 4 The null hypothesis for a particular experiment is that the mean test score is 20. If the 95% confidence interval is (18, 24), can you reject the null hypothesis? No. You cannot reject the null hypothesis because the confidence interval shows that 20 is a plausible value of the population parameter. 8
9. 9. Example 5 Which of these 95% confidence intervals represent samples that are significantly different from zero? Select all that apply. a) (4.6, 1.8) b) (0.2, 8.1) c) (5.1, 6.7) d) (3.0, 10.9) 9
10. 10. Example 6 True or false? If a 95% confidence interval contains 0, so will the 99% confidence interval. True. The 99% confidence interval contains all of the values that the 95% confidence interval has, but it extends farther at both ends and has other values, too. If something is not significant at the .05 level, it is also non-significant at the .01 level. 10
11. 11. Example 7 A researcher hypothesizes that the lowering in cholesterol associated with weight loss is really due to exercise. To test this, the researcher carefully controls for exercise while comparing the cholesterol levels of a group of subjects who lose weight by dieting with a control group that does not diet. The difference between groups in cholesterol is not significant. Can the researcher claim that weight loss has no effect? 11
12. 12. Problems with Significance Failing to reject the null hypothesis can mean either that (1) the null hypothesis is true, or (2) the alternative was actually true but there just wasn't enough evidence (a Type 2 Error). However, rejecting the null hypothesis can also be problematic. 12
13. 13. Example 8 Each graph below shows the difference of two proportions. Which one shows a statistically significant difference? 13
14. 14. Example 9 Now using the p-values, which one shows a statistically significant difference? What is going on? 14 p-value = 0.4902 p-value = 4.79 E 6
15. 15. Example 10 Now consider the sample sizes. How does that explain the p-values? 15 p-value = 0.4902 p-value = 4.79 E 6 A B Total Success 3 4 7 Failure 2 1 3 Total 5 5 10 A B Total Success 40,000 39,000 79,000 Failure 60,000 61,000 121,000 Total 100,000 100,000 200,000
16. 16. Problems with Significance One problem with hypothesis tests is that samples that are too large will tend to reject the null hypothesis regardless of any effect. The solution is that when you do reject the null hypothesis, you should also consider the effect size. 16
17. 17. Effect Size In most of the hypothesis-testing situations, we are interested in comparing a population mean or proportion to a specific null value. In many research situations, we would like to know something about the magnitude of the comparison. The test statistic and p-value for a test are not useful for this purpose because they depend on the size of the sample. 17
18. 18. Effect Size Statistical Significance: Measures the likelihood you could have gotten your results by random chance. P-values and confidence intervals are considered. Practical Significance: Measures the likelihood that the truth differs by chance. Effect size is considered, removing sample size from calculations. 18
19. 19. Cohen's d A common effect-size measure is Compare this to the test statistic: point estimate null value Cohen's standard deviation d 19 point estimate null value Test Statistic standard error Keep your final answer positive.
20. 20. Cohen's d The following table is somewhat arbitrary and should only be used as a guideline of the effect size. Effect Size Magnitude Interpretation Small 0.0 0.1 Not obvious without statistics Modest 0.1 0.3 Obvious only to very careful observers Moderate 0.3 0.5 Obvious to careful observers Large > 0.5 Obvious to most observers 20
21. 21. Example 11 Compare the significance and effect size for each difference. 21 p-value = 0.4902 p-value = 4.79 E 6 d = 0.0205d = 0.4364
22. 22. Hypotheses Testing Steps 1) State the null and alternate hypotheses (in symbols) 2) Choose the significance level (default is = .05) 3) Choose the test and check the assumptions 4) Calculate the test statistic 5) Calculate the p-value 6) Compare the p-value to alpha 7) Write the decision (reject or fail to reject null) 8) Write a meaningful conclusion about the alternate 9) If null is rejected, then check the effect size 22
23. 23. Example 12 Calculate and interpret the test statistic and effect size for each test. a) H0: = 20 vs. Ha: 20; = 19.8, SD = 1.5, SE = 0.25 b) H0: p = 0.1 vs. Ha: p 0.1; = 0.15, SD = 0.357, SE = 0.0595 x 23 p
24. 24. Example 13 In a recent year, of the 109,857 arrests for Federal offenses, 29.1% were for drug offenses (based on data from the U.S. Department of Justice). Test the claim that the drug offense rate is equal to 30%. How can the result be explained, given that 29.1% appears to be so close to 30%? Use = 0.291, SD = 0.454, and SE = 0.00137. 24 p
25. 25. Example 14 USA Today ran a report about a University of North Carolina poll of 1248 adults from the Southern United States. It was reported that 8% of those surveyed believe that Elvis Presley still lives. The article began with the claim that "almost 1 out of 10" Southerners still thinks Elvis is alive. Test the claim that the true percentage is less than 10%. Based on the result, determine whether the 8% sample result justifies the phrase "almost 1 out of 10." Use = 0.08, SD = 0.271, and SE = 0.00768. 25 p