Chapter 7 Testing Differences between Means

1. Chapter 7 Testing Differences between Means

2. Testing Differences Between Means • Establish hypothesis about populations, collect sample data, and see how likely the sample results are, given the hypothesis. • Example: Memory enhancement • N = 10

3. Now suppose instead that the following sets of scores produced the two sample means of 82 and 77.

4. The Null Hypothesis • No difference between means • An obtained difference between two sample means does not represent a true difference between their population means • Mean of the first population = mean of the second population • Retain or reject the null hypothesis • Null hypothesis shown as H0:

5. The Research Hypothesis • Differences between groups, whether expected on theoretical or empirical grounds, often provide the rationale for research • Mean of the first population does not equal the mean of the second population • If we reject the null hypothesis, we automatically accept the research hypothesis that a true population difference does exist. • Different means Research hypothesis shown as H1:

6. Levels of Significance • To establish whether our obtained sample difference is statistically significant – the result of a real population difference and not just sampling error – it is customary to set up a level of significance • Denoted by the Greek letter alpha (α) • The alpha value is the level of probability at which the null hypothesis can be rejected with confidence and the research hypothesis can be accepted with confidence.

7. Type I and II Errors DECISION Retain Null Reject Null Null is true REALITY Null is false

8. Choosing a Level of Significance • Suppose for example that a researcher were doing research on gender differences in sentence length for first time drug offenses for a random sample of males and females. What would be worse? Type I error or Type II error? • Suppose that a researcher is testing the effects of marijuana smoking on SAT performance, and he compares a sample of smokers with a sample of nonsmokers. What would be worse? Type I error or Type II error?

9. What is the Difference Between P and Alpha? • The difference between P and alpha can be a bit confusing

10. Standard Error of the Difference between Means • Standard deviation of the distribution of differences can be estimated. • The standard error of the differences between means is shown as:

11. Testing the Difference between Means • Why use t instead of z? • Test differences between means using t: • This is referred to as our T computed

12. Comparing our T value • Using Table C, we find our T critical value. • To calculate the degrees of freedom (df) when testing the difference between means we use the following formula • df = N1 + N2 – 2 • Alpha value is given (.05 or .01) • If T computed > T critical, reject null • If T computed < T critical, accept null

13. Testing the Difference between Means • Suppose that we obtained the following data for a sample of 25 liberals and 35 conservatives on the permissiveness scale. • Calculate the estimate of the standard error of the differences between means. • Then, translate the difference between sample means into a t ratio.

14. Continued. If necessary, find the mean and standard deviation first. Otherwise: • Step 1: Find the standard error of the difference between means. • Step 2: Compute the t ratio. • Step 3: Determine the critical value for t. • Step 4: Compare the calculated and table t values.

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16. Comparing the Same Sample Measured Twice • So far, we have discussed making comparisons between two independently drawn samples • Before-after or panel design: the case of a single sample measured at two different points in time (time 1 vs. time 2) • For example, a polling organization might interview the same 1,000 Americans both in 1995 and 2000 in order to measure their change in attitude over time. • Numerous uses for this type of test

17. Testing the Difference Between Means for the Same Sample Measured Twice • To obtain the standard error of the difference between means use the following formula: • Where: • SD = Standard deviation of the distribution of before-after difference scores.

18. Finding the t ratio Computed T ratio: Critical T:df = N – 1α = .05 or .01 Use Table C Compare the computed T with the critical T.If |T| > critical T, reject null hypothesis. If |T| < critical T, retain null hypothesis.

19. Test of Difference between Means for Same Sample Measured Twice • Suppose that several individuals have been forced by a city government to relocate their homes to make way for highway construction. • As researchers, we are interested in determining the impact of forced residential mobility on feelings of neighborliness. • What would the null and research hypotheses state? • We interview a random sample of 6 individuals about their neighbors both before and after they are forced to move.

20. Their Scores

21. Two Sample Test of Proportions • As in Chapter 6, we are interested in testing the difference between two groups measured in proportions. • Males/Females, Blacks/Whites, Liberals/Conservatives, Violent/Nonviolent criminals, Adult/Juvenile offenders, etc • Use Z scores for critical values • When alpha = .05, Z score of 1.96 is used • When alpha = .01, Z score of 2.58 is used • Use for stating the null/research hypotheses

22. Two Sample Test of Proportions Formulas Step 1: Find P* (combined sample proportion). P* Step 2: Standard error of the difference of proportions. Sp-p = Step 3: Find the Z computed score. z = Step 4: Compare Z computed with Z critical & interpret.

23. Two Sample Test Example A criminal justice researcher is interested in marijuana usage and driving while high of upper level undergraduates in her particular school. After taking a random sample of 300 students, she discards any surveys of students who have not smoked marijuana. She is left with the following data: Test the research hypothesis at the alpha level of .05. What do your results indicate?

24. One-Tailed Tests • A one-tailed test rejects the null hypothesis at only one tail of the sampling distribution. • It should be emphasized, however, that the only changes are in the way the hypotheses are stated and the place where the t table is entered. • Used when the researcher anticipates the direction of change.

25. Requirments for Testing the Differences between Means • A comparison between two means • Interval data • Random sampling • A normal distribution • Equal population variances