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Chapter 10

Simple Regression

And Correlation

Page 184-185 - Testing for Significance

There is an alternate and potentially simpler procedure for testing whether a given correlation coefficient is significant. This is presented in detail bellow, in the discussion of Practice Exercise 5.

Page 191 - 194 - Part b of each Practice Exercise

Part b of each problem asks you to "Calculate the slope (a) and the intercept (b) for this data."

This ends up being quite confusing since the statistical symbol for slope is b and the symbol for the y-intercept is a.

This should be read as "Calculate 1) the slope (b) and 2) the intercept (a) for this data.

Page 194 - Practice Exercise 5

This problem, though not impossible, is unnecessarily difficult when using the r to t conversion procedure for testing the significance of r as outlined in on pages 184-185 (Testing for Significance). Because the conversion formula includes sample size as part of its calculation, a separate calculation would have to be computed for each change in the sample size.

However, in the Instructors Guide, and again here, we have provided an alternate table for testing the significance of r. This table is used just like the t table (in fact it is computed based on the table of critical values for t). This table provides you with the critical values of r that you have to match or exceed for different levels of significance (both one tailed or two tailed) and for different degrees of freedom (n-2).

This table can be used to solve Practice Exercise 5. Simply find the .05 column (for two tailed tests) and move down the column until you find the first critical value that matches or falls bellow the r value that you are working with. Then look to the far left in that row to find the degrees of freedom. Add 2 to the degrees of freedom, and that will tell you the number of subjects that are need in order for the obtained r to be significant.

For Example: Assume that we have data on Shoe Size and Attitude Toward Cheezy Poofs for 24 people. After computing r we get an r obtained of .3620. For df = 22, at .05 alpha level (two-tailed) the critical value of r is .404. We did not meet or beat this value, so we reject the Alternative/Research Hypothesis and Fail to reject the Null Hypothesis. The relationship between Shoe Size and Attitude Toward Cheezy Poofs is not significant. However, we can see that there is some kind of relationship; it is just not large enough for us to be 95% confident that it is a real relationship given the number of people in our sample. How many people would we need to add for this relationship to be considered significant; for us to be 95% confident? Looking at the .05 (Two-Tailed) column, the first critical value for r that we meet or beat is .361, which is the critical value for df = 28. Thus we need 30 people (an additional 6 people) in order for the correlation between shoe size and attitude toward cheezy poofs to be considered significant. This assumes that the 6 additional people came from the same population as the original sample and that the obtained correlation does not appreciably change.

Table of Critical Values for Pearson’s r (printable pdf)

Adapted from Appendix 2 (Critical Values of t) using the square root of [t²/(t² _­­+ df)]

Note: Critical Values for infinite degrees of freedom actually calculated at df = 500.