1. t tests comparing two means t tests comparing two means

2. Overall Purpose A t-test is used to compare two average scores. A t-test is used to compare two average scores. Sample data are used to answer a question about population means. Sample data are used to answer a question about population means. The population means and standard deviations are not known. The population means and standard deviations are not known.

3. The Three Types There are three ways to use a There are three ways to use a t-test in a comparative educational study: 1. To compare two groups measured at one time. 1. To compare two groups measured at one time. Independent t-test Independent t-test

4. The Three Types 2. To compare a sample to a population, or one group 2. To compare a sample to a population, or one group measured at one point in time. measured at one point in time. One sample t-test One sample t-test

5. The Three Types 3. To compare one group to itself over time, or one group measured at two times. 3. To compare one group to itself over time, or one group measured at two times. Dependent t-test Dependent t-test

6. Assumptions Normality Normality Homogeneity of Variance Homogeneity of Variance Independence of Observations Independence of Observations Random Sampling Random Sampling Population Variance Not Known Population Variance Not Known

7. Additional Considerations Confidence intervals can be used to test the same hypotheses. Confidence intervals can be used to test the same hypotheses. There is a unique critical t value for each degrees of freedom condition. There is a unique critical t value for each degrees of freedom condition.

8. Additional Considerations Random assignment is needed to make causal inferences when using the independent t-test. Random assignment is needed to make causal inferences when using the independent t-test. If intact groups are compared, examine differences on potential confounding variables. If intact groups are compared, examine differences on potential confounding variables.

9. Additional Considerations The z value of 1.96 serves as a rough guideline for evaluating a The z value of 1.96 serves as a rough guideline for evaluating a t value. t value. It means that the amount of difference in the means is approximately twice as large as expected due to sampling error alone. It means that the amount of difference in the means is approximately twice as large as expected due to sampling error alone. Interpret the p value. Interpret the p value.

10. Statistical Significance How do you know when there is a statistically significant difference between the average scores you are comparing? How do you know when there is a statistically significant difference between the average scores you are comparing?

11. Statistical Significance When the p value is less than alpha, usually set at.05. When the p value is less than alpha, usually set at.05. What does a small p value mean? What does a small p value mean?

12. Statistical Significance If the two population means are equal, your sample data can still show a difference due to sampling error. If the two population means are equal, your sample data can still show a difference due to sampling error. The p value indicates the probability of results such as those obtained, or larger, given that the null hypothesis is true and only sampling error has lead to the observed difference. The p value indicates the probability of results such as those obtained, or larger, given that the null hypothesis is true and only sampling error has lead to the observed difference.

13. Statistical Significance You have to decide which is a more reasonable conclusion: You have to decide which is a more reasonable conclusion: There is a real difference between the population means. There is a real difference between the population means.Or The observed difference is due to sampling error. The observed difference is due to sampling error.

14. Statistical Significance We call these conclusions: We call these conclusions: Rejecting the null hypothesis. Rejecting the null hypothesis. Failing to reject the null hypothesis. Failing to reject the null hypothesis.

15. Statistical Significance If the p value is small, less than alpha (typically set at.05), then we conclude that the observed sample difference is unlikely to be the result of sampling error. If the p value is small, less than alpha (typically set at.05), then we conclude that the observed sample difference is unlikely to be the result of sampling error.

16. Statistical Significance If the p value is large, greater than or equal to alpha (typically set at.05), then we conclude that the difference you observed could have occurred by sampling error, even when the null hypothesis is true. If the p value is large, greater than or equal to alpha (typically set at.05), then we conclude that the difference you observed could have occurred by sampling error, even when the null hypothesis is true.

17. Hypotheses Hypotheses for the Independent t-test Hypotheses for the Independent t-test Null Hypothesis:  1 =  2 or  1 -  2 = 0  1 =  2 or  1 -  2 = 0 Directional Alternative Hypothesis:  1 >  2 or  1 -  2 > 0  1 >  2 or  1 -  2 > 0 Non-directional Alternative Hypothesis:  1 =/=  2 or  1 -  2 =/= 0  1 =/=  2 or  1 -  2 =/= 0where:  1 = population mean for group one  1 = population mean for group one  2 = population mean for group two  2 = population mean for group two

18. Hypotheses Hypotheses for the One Sample t-test Hypotheses for the One Sample t-test Null Hypothesis:  =  0 or  -  0 = 0  =  0 or  -  0 = 0 Directional Alternative Hypothesis:  >  0 or  -  0 > 0  >  0 or  -  0 > 0 Non-directional Alternative Hypothesis:  =/=  0 or  -  0 =/= 0  =/=  0 or  -  0 =/= 0where:  = population mean for group of interest (local population)  = population mean for group of interest (local population)  0 = population mean for comparison (national norm)  0 = population mean for comparison (national norm)

19. Hypotheses Hypotheses for the Dependent t-test Hypotheses for the Dependent t-test Null Hypothesis: d = 0 or  1 -  2 = 0 d = 0 or  1 -  2 = 0 Directional Alternative Hypothesis: d > 0 or  1 -  2 > 0 d > 0 or  1 -  2 > 0 Non-directional Alternative Hypothesis: d =/= 0 or  1 -  2 =/= 0 d =/= 0 or  1 -  2 =/= 0where:  1 = population mean for time one  1 = population mean for time one  2 = population mean for time two  2 = population mean for time two d = the average difference between time on and time two. d = the average difference between time on and time two.

20. Example Our research design: Our research design: We could compare “leavers” and “stayers” on their reported Classroom Demands, Resources, and Stress. We could compare “leavers” and “stayers” on their reported Classroom Demands, Resources, and Stress.