476 CHAPTER 9 Correlation and Regression Using a Table to Test a Population Correlation Coefficient R Once you have calculated r, the sample correlation coefficient, you will want to determine whether there is enough evidence to decide that the population correlation coefficient r is significant. In other words, based on a few pairs of data, can you make an inference about the population of all such data pairs? Remember that you are using sample data to make a decision about population data, so it is always possible that your inference may be wrong. In correlation studies, the small percentage of times when you decide that the correlation is significant when it is really not is called the level of significance. It is typically set at a = 0.01 or 0.05. When a = 0.05, you will probably decide that the population correlation coefficient is significant when it is really not 5% of the time. (Of course, 95% of the time, you will correctly determine that a correlation coefficient is significant.) When a = 0.01, you will make this type of error only 1% of the time. When using a lower level of significance, however, you may fail to identify some significant correlations. For a correlation coefficient to be significant, its absolute value must be close to 1. To determine whether the population correlation coefficient r is significant, use the critical values given in Table 11 in Appendix B. A portion of the table is shown below. If 0 r0 is greater than the critical value, then there is enough evidence to decide that the correlation is significant. Otherwise, there is not enough evidence to say that the correlation is significant. For instance, to determine whether r is significant for five pairs of data 1n = 52 at a level of significance of a = 0.01, you need to compare 0 r0 with a critical value of 0.959, as shown in the table. n = 0.05 4 0.950 6 0.811 = 0.01 0.990 0.917 a a 5 0.878 0.959 Number n of pairs of data in sample Critical values for = 0.05 and = 0.01 α α If 0 r0 7 0.959, then the correlation is significant. Otherwise, there is not enough evidence to conclude that the correlation is significant. Here are the guidelines for this process. Using Table 11 for the Correlation Coefficient R In Words In Symbols 1. Determine the number of pairs Determine n. of data in the sample. 2. Specify the level of significance. Identify a. 3. Find the critical value. Use Table 11 in Appendix B. 4. Decide whether the correlation If 0 r0 is greater than the critical is significant. value, then the correlation is significant. Otherwise, there is not enough evidence to conclude that the correlation is significant. 5. Interpret the decision in the context of the original claim. GUIDELINES Study Tip The symbol 0 r 0 represents the absolute value of r. Recall that the absolute value of a number is its value, disregarding its sign. For example, 0 30 = 3 and 0 -70 = 7. Study Tip The level of significance is denoted by a, the lowercase Greek letter alpha. Study Tip If you determine that the linear correlation is significant, then you will be able to proceed to write the equation for the line that best describes the data. This line, called the regression line, can be used to predict the value of y when given a value of x. You will learn how to write this equation in the next section.
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