Chapter Eleven

1. Chapter Eleven Performing the One-Sample t-Test and Testing Correlation

2. More Statistical Notation Recall the formula for the estimated population standard deviation Chapter 11 - 2

3. Use the z-test when is known Use the t-test when is estimated by calculating Using the t-Test Chapter 11 - 3

4. Performing the One-Sample t-Test Chapter 11 - 4

5. Setting Up the Statistical Test Set up the statistical hypotheses (H0 and Ha). These are done in precisely the same fashion as in the z-test. Select alpha Check the assumptions for a t-test Chapter 11 - 5

6. You have one random sample of interval or ratio scores The raw score population forms a normal distribution The standard deviation of the raw score population is estimated by computing Assumptions for a t-Test Chapter 11 - 6

7. Computational Formula for the t-Test First, compute the estimated standard error of the mean Chapter 11 - 7

8. Computational Formula for the t-Test Then, compute the one-sample t statistic: Chapter 11 - 8

9. The t-Distribution The t-distribution is the distribution of all possible values of t computed for random sample means selected from the raw score population described by H0 Chapter 11 - 9

10. Comparison of Two t-distributions Based on Different Sample Ns Chapter 11 - 10

11. The quantity N - 1 is called the degrees of freedom We obtain the appropriate value of tcritfrom the t-tables using both the appropriate a and df Degrees of Freedom Chapter 11 - 11

12. Two-Tailed t-Distribution Chapter 11 - 12

13. Estimating the Population Mean by Computing a Confidence Interval Chapter 11 - 13

14. Estimating m There are two ways to estimate the population mean m Point estimation in which we describe a point on the variable at which the m is expected to fall Interval estimation in which we specify an interval (or range of values) within which we expect the m to fall Chapter 11 - 14

15. Confidence Intervals We perform interval estimation by creating a confidence interval The confidence interval for a single m describes an interval containing values of m Chapter 11 - 15

16. Significance Tests for Correlation Coefficients Chapter 11 - 16

17. The Pearson Correlation Coefficient The Pearson correlation coefficient (r) is used to describe the relationship in a sample Ultimately we want to describe the relationship in the population For any correlation coefficient you compute, you must decide if it is significant Chapter 11 - 17

18. The Pearson Correlation Coefficient The symbol for the Person correlation coefficient in the population is r Chapter 11 - 18

19. Hypotheses Two-tailed test H0: r = 0 Ha: r ≠ 0 One-tailed test Predicting positive − Predicting negative correlation correlation H0: r ≤ 0 H0: r ≥ 0 Ha: r > 0 Ha: r< 0 Chapter 11 - 19

20. Scatterplot of a Population for Which r = 0 Chapter 11 - 20

21. Assumptions for the Pearson r There is a random sample of X and Y pairs and each variable is an interval or ratio variable. The X scores and Y the scores each represent a normal distribution. Further, they represent a bivariate normal distribution. The null hypothesis is there is zero correlation in the population. Chapter 11 - 21

22. Sampling Distribution The sampling distribution of ris a frequency distribution showing all possible values of r that can occur when samples of size N are drawn from a population where r is zero. Chapter 11 - 22

23. Degrees of Freedom The degrees of freedom for the significance test of a Pearson correlation coefficient are N - 2 N is the number of pairs of scores Chapter 11 - 23

24. Interpreting the Results • If the Pearson r is significant, compute • the regression equation and • the proportion of variance accounted for (r2) • It is the r2 (not the test of significance) that indicates the importance of the relationship Chapter 11 - 24

25. Testing the Spearman rs Testing the Spearman rs requires a random sample of pairs of ranked (ordinal) scores Use the critical values of the Spearman rank-order correlation coefficient for either a one-tailed or a two-tailed test The critical value is obtained using N, the number of pairs of scores in the sample Chapter 11 - 25

26. Maximizing the Power of a Statistical Test Chapter 11 - 26

27. Maximizing the Power of the t-Test Larger differences produced by changing the independent variable increase power Smaller variability in the raw scores increases power A larger N increases power Chapter 11 - 27

28. Maximizing the Power of aCorrelation Coefficient Avoiding a restricted range increases power Minimizing the variability of the Y scores at each X increases power Increasing N increases power Chapter 11 - 28

29. Example 1 Use the following data set and conduct a two-tailed t-test to determine if m = 12 Chapter 11 - 29

30. Example 1 H0: m = 12; Ha: m ≠ 12 Choose a = 0.05 Reject H0 if tobt > +2.110 or if tobt < -2.110 Chapter 11 - 30

31. Example 2 For the following data set, determine if the Pearson correlation coefficient is significant. Chapter 11 - 31

32. Example 2 From chapter 7, we know that r = -0.88 Using a = 0.05 and a two-tailed test, rcrit = 0.811. Therefore, we will reject H0 if robt > 0.811 or if robt < -0.811 Since robt = -0.88, we reject H0 We conclude this correlation coefficient is significantly different from 0 Chapter 11 - 32

33. Key Terms • point estimation • sampling distribution of r • sampling distribution of rs • t-distribution confidence interval for a single m estimated standard error of the mean interval estimation margin of error one-sample t-test Chapter 11 - 33