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

2. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 2 More Statistical Notation Recall the formula for the estimated population standard deviation.

3. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 3 Use the z-test when is known. Use the t-test when is estimated by calculating. Using the t-Test

4. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 4 Performing the One-Sample t-Test

5. Copyright © Houghton Mifflin Company. All rights reserved.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. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 6 Assumptions for a t-Test 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

7. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 7 Computational Formula for the t-Test

8. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 8 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

9. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 9 Comparison of Two t-distributions Based on Different Sample N s

10. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 10 The quantity N - 1 is called the degrees of freedom Since it is this value that is used to compute, it is the degrees of freedom (df) that determines how consistently estimates the true We obtain the appropriate value of t crit from the t-tables using both the appropriate  and df Degrees of Freedom

11. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 11 Two-Tailed t-Distribution [Insert Figure 11.3 here.]

12. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 12 Estimating the Population Mean by Computing a Confidence Interval

13. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 13 Estimating  There are two ways to estimate the population mean  Point estimation in which we describe a point on the variable at which the population mean is expected to fall Interval estimation in which we specify an interval (or range of values) within which we expect the population parameter to fall

14. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 14 Confidence Intervals We perform interval estimation by creating a confidence interval The confidence interval for a single  describes an interval containing values of 

15. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 15 Significance Tests for Correlation Coefficients

16. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 16 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

17. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 17 Hypotheses Two-tailed test –H 0 :  = 0 –H a :  ≠ 0 One-tailed test –Predicting positive Predicting negative correlation correlation H 0 :  ≤ 0 H 0 :  ≥ 0 H a :  > 0 H a :  < 0

18. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 18 Scatterplot of a Population for Which  = 0

19. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 19 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 Y scores and the X scores each represent a normal distribution. Further, they represent a bivariate normal distribution. 3.The null hypothesis is that in the population there is zero correlation

20. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 20 Sampling Distribution The sampling distribution of a correlation coefficient is a frequency distribution showing all possible values of the coefficient that can occur when samples of size N are drawn from a population where  is zero

21. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 21 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.

22. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 22 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.

23. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 23 Maximizing the Power of a Statistical Test

24. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 24 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

25. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 25 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

26. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 26 14 13151115 131012131413 14151714 15 Example 1 Use the following data set and conduct a two-tailed t-test to determine if  = 12

27. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 27 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

28. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 28 X Y 15 22 36 44 53 61 Example 2 For the following data set, determine if the Pearson correlation coefficient is significant.

29. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 11 - 29 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 that this correlation coefficient is significantly different from 0