Copyright © Cengage Learning. All rights reserved. 13 Linear Correlation and Regression Analysis

Copyright © Cengage Learning. All rights reserved Inferences about the Linear Correlation Coefficient

3 Inferences about the Linear Correlation Coefficient Assumptions for inferences about the linear correlation coefficient The set of (x, y) ordered pairs forms a random sample, and the y values at each x have a normal distribution. Inferences use the t-distribution with n – 2 degrees of freedom. Caution Inferences about the linear correlation coefficient are about the pattern of behavior of the two variables involved and the usefulness of one variable in predicting the other. Significance of the linear correlation coefficient does not mean that you have established a cause-and-effect relationship. Cause and effect is a separate issue.

4 Confidence Interval Procedure

5 As with other parameters, a confidence interval may be used to estimate the value of , the linear correlation coefficient of the population.

6 Example 2 – Constructing a Confidence Interval for the Population Correlation Coefficient A random sample of 15 ordered pairs of data has a calculated r value of Find the 95% confidence interval for , the population linear correlation coefficient. Solution: Step 1 Parameter of interest: The linear correlation coefficient for the population,  Step 2 a. Assumptions: The ordered pairs form a random sample, and we will assume that the y values at each x have a normal distribution. b. Formula: The calculated linear correlation coefficient, r

7 Example 2 – Solution c. Level of confidence: 1 –  = 0.95 Step 3 Sample information: n = 15 and r = 0.35 Step 4 Confidence interval: The confidence interval is read from Table 10 in Appendix B. Find r = 0.35 at the bottom of Table 10. (See the arrow on Figure 13.5.) cont’d Using Table 10 of Appendix B, Confidence Belts for the Correlation Coefficient Figure 13.5

8 Example 2 – Solution Visualize a vertical line drawn through that point. Find the two points where the belts marked for the correct sample size cross the vertical line. The sample size is 15. These two points are circled in Figure Now look horizontally from the two circled points to the vertical scale on the left and read the confidence interval. The values are –0.20 and Step 5 Confidence interval: The 95% confidence interval for , the population coefficient of linear correlation, is –0.20 to cont’d

9 Hypothesis-Testing Procedure

10 Hypothesis-Testing Procedure The null hypothesis is: The two variables are linearly unrelated (  = 0), where  is the linear correlation coefficient for the population. The alternative hypothesis may be either one-tailed or two-tailed. Most frequently it is two-tailed,  ≠ 0. However, when we suspect that there is only a positive or only a negative correlation, we should use a one-tailed test. The alternative hypothesis of a one-tailed test is  > 0 or  < 0.

11 Hypothesis-Testing Procedure The area that represents the p-value or the critical region for the test is on the right when a positive correlation is expected and on the left when a negative correlation is expected. The test statistic used to test the null hypothesis is the calculated value of r from the sample data.

12 Example 3 – Two-tailed Hypothesis Test In a study of 15 randomly selected ordered pairs, r = Is this linear correlation coefficient significantly different from zero at the 0.02 level of significance? Solution: Step 1 a. Parameter of interest: The linear correlation coefficient for the population,  b. Statement of hypotheses: H o :  = 0 H a :  ≠ 0

13 Example 3 – Solution Step 2 a. Assumptions: The ordered pairs form a random sample, and we will assume that the y values at each x have a normal distribution. b. Test statistic:,formula (13.3), with df = n – 2 = 15 – 2 = 13 c. Level of significance:  = 0.02 (given in the statement of the problem) Step 3 a. Sample information: n = 15 and r = b. Value of the test statistic: The calculated sample linear correlation coefficient is the test statistic: = cont’d

14 Example 3 – Solution Step 4 Probability Distribution: p-Value: a. Use both tails because H a expresses concern for values related to “different from.” P = P (r 0.548) = 2  P (r > 0.548) with df = 13, as shown in the figure. cont’d

15 Example 3 – Solution Use Table 11 (Appendix B) to place bounds on the p-value: 0.02 < P < b. The p-value is not smaller than the level of significance, . Classical: a. The critical region is both tails because H a expresses concern for values related to “different from.” cont’d

16 Example 3 – Solution The critical value is found at the intersection of the df = 13 row and the two-tailed 0.02 column of Table 11: b. is not in the critical region, as shown in red in the figure. cont’d

17 Example 3 – Solution Step 5 a. Decision: Fail to reject H o. b. Conclusion: At the 0.02 level of significance, we have failed to show that x and y are correlated. cont’d

18 Hypothesis-Testing Procedure Calculating the p-Value Use Table 11 in Appendix B to “place bounds” on the p-value. By inspecting the df = 13 row of Table 11, you can determine an interval within which the p-value lies. Locate along the row labeled df = 13 If is not listed, locate the two table values it falls between, and read the bounds for the p-value from the top of the table. In this case, = is between and 0.592; therefore, P is between 0.02 and 0.05.

19 Hypothesis-Testing Procedure Table 11 shows only two-tailed values. When the alternative hypothesis is two-tailed, the bounds for the p-value are read directly from the table. Note When H a is one-tailed, divide the column headings by 2 to place bounds on the p-value.

20 Hypothesis-Testing Procedure Use Table 11 in Appendix B to find the critical values. The critical value is at the intersection of the df = 13 and the two-tailed  = 0.02 column. Table 11 shows only two-tailed values. Since the alternative hypothesis is two-tailed, the critical values are read directly from the table. Note When H a is one-tailed, divide the column headings by 2.