1. 6/4/2016Slide 1 The one sample t-test compares two values for the population mean of a single variable. The two-sample t-test of population means (aka independent samples t-test) compares the population means for two groups of subjects on a single variable. The null hypothesis for this test is: there is no difference between the population mean of the variable for one group of subjects and the population mean of the same variable for a second group of subjects. In addition to our concern with the assumption of normality for each group and the number of cases in each group if we are to apply the Central Limit Theorem, this test also requires us to examine the spread or dispersion of both groups so that the measure of standard error used in the t-test fairly represents both group. The equality of the spread across groups is tested with the Levene test of homogeneity of variance. The Levene test of homogeneity of variance tests the null hypothesis that the variances of the two groups are equal versus the alternative hypothesis that the variance of the first group is different from the variance of the second group. Failure to reject the null supports the assumption of homogeneity, and is the desired outcome. However, the failure to satisfy the assumption of homogeneity in the independent samples t-test is easily corrected by using an alternative formula for computing the standard error of the sampling distribution.

2. 6/4/2016Slide 2 The authors of our text suggest we always use the formula that assumes the homogenity of variance assumption is violated. If we use this version of the statistic and the variances are in fact equal, the results of the test are comparable to what we would obtain using the formula for equal variances. We will not take the authors advice, but will instead pay attention to the Levene test as an example of hypothesis tests of diagnostic statistics. In these t-tests and many other tests, we will state the degrees of freedom (df) in our report of the statistical test. Degrees of freedom is generally based on the sample size and the number of statistics that are computed. Degrees of freedom start out equal to the number of cases in our sample, and decrease for each statistic that we compute. For example, in computing the variance or standard deviation, the total sum of squares is divided by the number of cases – 1, where the minus 1 is the 1 degree of freedom that is used when we computed the mean in order to compute the deviations from the mean. After computing the mean, n – 1 of the scores in the distribution are still free do vary, so we say we have n – 1 degrees of freedom left. SPSS generally reports the degrees of freedom in most tests, and we can use the number reported in our presentation of statistical results.

3. 6/4/2016Slide 3 This is the template for independent samples t-test problems with correct answers filled in.

4. 6/4/2016Slide 4 The first two blanks want you to enter the number in each of the groups. The number should be entered as an integer without decimal fractions, but it will not be marked incorrect if decimals are included. To obtain the number in each group, as well as the statistics needed, we run the Independent Samples T Test.

5. 6/4/2016Slide 5 To produce the Independent- Samples T-Test, select the Compare Means > Independent Samples T Test command from the Analyze menu. If you are doing these problems right after doing the one-sample t-test problems, close and re-open the data set to make sure you are including all cases.

6. 6/4/2016Slide 6 Second, click on the right arrow button to move prison to the Test Variable(s) list box. First, highlight the dependent variable prison in the list of variables.

7. 6/4/2016Slide 7 Second, click on the right arrow button to move freeMove to the Grouping Variable text box. First, highlight the independent variable freeMove in the list of variables.

8. 6/4/2016Slide 8 Next, click on the Define Groups button to enter the numeric codes for the groups. When the independent variable is entered, SPSS appends two questions marks that it wants replaced with the code numbers for the groups.

9. 6/4/2016Slide 9 First, enter 0 for restricted as Group 1. Second, enter 1 for generally unrestricted as Group 2. Third, click on the Continue button to close the dialog box. If I did not remember the code numbers for the freedom of movement variable, I would look them up in the Variable View of the SPSS Data Editor.

10. 6/4/2016Slide 10 Third, click on the OK button to produce the output. SPSS replaces the question marks with the codes I entered.

11. 6/4/2016Slide 11 The number of cases in each group is listed in the table of group statistics. Group 1 (restricted) had 64 cases, while group 2 (generally unrestricted) had 107 cases.

12. 6/4/2016Slide 12 The next item is a drop-down menu of choices to summarize our assessment of the normality of the distribution of the dependent variable. The criteria for normality are stated in the notes.

13. 6/4/2016Slide 13 The distribution is not nearly normal for all three criteria: skewness, kurtosis, and number of outliers. To continue with the problem, we will need to rely on the central limit theorem to satisfy the assumption of normality for the sampling distribution. The criteria for the central limit theorem for the independent samples t-test is listed in the notes.

14. 6/4/2016Slide 14 The central limit theorem requires that we have 40 cases in each group. We have 64 and 107, so the distribution of sample differences will be normally distributed even though the sample distribution for the variable is not.

15. 6/4/2016Slide 15 The next blanks expect us to enter the value of the F ratio and p-value for the Levene test of homogeneity of variance. The Levene test of homogeneity of variance tests the null hypothesis that the variance of the two groups is equal versus the alternative hypothesis that the variance of the first group is different from the variance of the second group. Failure to reject the null supports the assumption of homogeneity, and is the desired outcome. However, the failure to satisfy the assumption of homogeneity in the independent samples t-test is easily corrected by using an alternative formula for computing the standard error of the sampling distribution.

16. 6/4/2016Slide 16 We transfer the statistics from the Levene test to the blanks in the problem, and select the “=“ sign for the comparison of the p- value.

17. 6/4/2016Slide 17 Since the p-value for the Levene test (.124) was greater than alpha, we fail to reject the null hypothesis of equal variance and conclude that there is not a significant violation.

18. 6/4/2016Slide 18 Since there was no significant violation, the variances were equal and we use the pooled variance formula (equal variances assumed in SPSS) for the t-test. Had there been a significant violation, the variances were not equal and we use the separate variance formula (equal variances not assumed in SPSS) for the t-test.

19. 6/4/2016Slide 19 The first sentence of the next paragraph compares the means and standard deviations of the groups.

20. 6/4/2016Slide 20 We transfer the mean and standard deviation for the restricted group from the Group Statistics table.

21. 6/4/2016Slide 21 We transfer the mean and standard deviation for the unrestricted group from the Group Statistics table.

22. 6/4/2016Slide 22 Since the mean of the restricted group was lower than the mean of the unrestricted group, we choose lower for the comparison.

23. 6/4/2016Slide 23 The second sentence in this paragraph addresses the statistical significance of the difference in group means. We are testing the null hypothesis that the population mean represented by group 1 is equal to the population mean represented by group 2, which can also be stated as: the population mean for group 2 minus the population mean for group 1 is equal to 0. The alternative hypothesis is that the population means represented by the two sample groups are not the same. This is a two-tailed hypothesis. One-tailed differences can also be tested, but they are not used in these problems.

24. 6/4/2016Slide 24 We transfer the results for the statistical test from the SPSS output to the narrative description in the APA format. We report the results in APA style, i.e. t(degrees of freedom) = t statistic, p = the p-value, followed by an indication that a one-tailed or two-tailed significance was interpreted.

25. 6/4/2016Slide 25 Since the p-value of.035 is less than the alpha of.05, we reject the null hypothesis and assert that the difference in the sample means was statistically different.

26. 6/4/2016Slide 26 Though it is redundant information, we will include the confidence interval in the problem narrative. It is redundant information because it is simply a different strategy for deciding whether or not to reject the null hypothesis.

27. 6/4/2016Slide 27 We enter the lower and upper bound for the 95% confidence interval from the SPSS output. The null hypothesis stated that the difference between the population means represented by the two sample means was 0. We reject the null hypothesis because 0 does not fall within the confidence interval.

28. 6/4/2016Slide 28 Finally, we select a summary statement for the test. If the test was not statistically significant, we would conclude that the groups had a similar prison population. If the test is statistically significant, we choose the conclusion that correctly states whether the first group was higher or lower than the second group.

29. 6/4/2016Slide 29 Since the t-test in this problem was statistically significant, and countries with restricted travel had a lower mean, we conclude that countries with restricted travel had a lower prison population per hundred thousand population.

30. 6/4/2016Slide 30 The green shading of the answers when we submit the problem indicate that our answers are correct.