1. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter Eleven Performing the One-Sample t-Test and Testing Correlation

2. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 2 More Statistical Notation Recall the formula for the estimated population standard deviation

3. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 3 Use the z-test when is known Use the t-test when is estimated by calculating Using the t-Test

4. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 4 Performing the One-Sample t-Test

5. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 5 Setting Up the Statistical Test 1.Set up the statistical hypotheses (H 0 and H a ). These are done in precisely the same fashion as in the z-test. 2.Select alpha 3.Check the assumptions for a t-test

6. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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 Chapter 11 - 6 Assumptions for a t-Test

7. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Computational Formula for the t-Test First, compute the estimated standard error of the mean Chapter 11 - 7

8. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Computational Formula for the t-Test Chapter 11 - 8 Then, compute the one-sample t statistic:

9. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 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 H 0

10. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 10 Comparison of Two t-distributions Based on Different Sample N s

11. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 11 The quantity N - 1 is called the degrees of freedom We obtain the appropriate value of t crit from the t-tables using both the appropriate  and df Degrees of Freedom

12. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 12 Two-Tailed t-Distribution

13. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 13 Estimating the Population Mean by Computing a Confidence Interval

14. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 14 Estimating  There are two ways to estimate the population mean  –Point estimation in which we describe a point on the variable at which the  is expected to fall –Interval estimation in which we specify an interval (or range of values) within which we expect the  to fall

15. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 15 Confidence Intervals We perform interval estimation by creating a confidence interval The confidence interval for a single  describes an interval containing values of 

16. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 16 Significance Tests for Correlation Coefficients

17. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 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

18. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. The Pearson Correlation Coefficient The symbol for the Person correlation coefficient in the population is  Chapter 11 - 18

19. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 19 Hypotheses Two-tailed test –H 0 :  = 0 –H a :  ≠ 0 One-tailed test –Predicting positive − Predicting negative correlation correlation H 0 :  ≤ 0H 0 :  ≥ 0 H a :  > 0H a :  < 0

20. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 20 Scatterplot of a Population for Which  = 0

21. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 21 Assumptions for the Pearson r 1.There is a random sample of X and Y pairs and each variable is an interval or ratio variable. 2.The X scores and Y the scores each represent a normal distribution. Further, they represent a bivariate normal distribution. 3.The null hypothesis is there is zero correlation in the population.

22. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 22 Sampling Distribution The sampling distribution of r is a frequency distribution showing all possible values of r that can occur when samples of size N are drawn from a population where  is zero.

23. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 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

24. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Interpreting the Results If the Pearson r is significant, compute –the regression equation and –the proportion of variance accounted for (r 2 ) It is the r 2 (not the test of significance) that indicates the importance of the relationship Chapter 11 - 24

25. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 25 Testing the Spearman r s Testing the Spearman r s 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

26. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 26 Maximizing the Power of a Statistical Test

27. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 27 Maximizing the Power of the t-Test 1.Larger differences produced by changing the independent variable increase power 2.Smaller variability in the raw scores increases power 3.A larger N increases power

28. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 28 Maximizing the Power of a Correlation Coefficient Avoiding a restricted range increases power Minimizing the variability of the Y scores at each X increases power Increasing N increases power

29. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 29 14 13151115 131012131413 14151714 15 Example 1 Use the following data set and conduct a two-tailed t-test to determine if  = 12

30. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 30 Example 1 H 0 :  = 12; H a :  ≠ 12 Choose  = 0.05 Reject H 0 if t obt > +2.110 or if t obt < -2.110

31. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 31 X Y 15 22 36 44 53 61 Example 2 For the following data set, determine if the Pearson correlation coefficient is significant.

32. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 11 - 32 Example 2 From chapter 7, we know that r = -0.88 Using  = 0.05 and a two-tailed test, r crit = 0.811. Therefore, we will reject H 0 if r obt > 0.811 or if r obt < -0.811 Since r obt = -0.88, we reject H 0 We conclude this correlation coefficient is significantly different from 0

33. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms confidence interval for a single  estimated standard error of the mean interval estimation margin of error one-sample t-test Chapter 11 - 33 point estimation sampling distribution of r sampling distribution of r s t-distribution