4.2 Testing the significance of the correlation coefficient  (Page 3/6)

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

Method 2: using a table of critical values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of $r$ is significant or not . Compare r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

Suppose you computed r = 0.801 using n = 10 data points. df = n - 2 = 10 - 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r <negative critical value or r >positive critical value, then r issignificant. Since r = 0.801 and 0.801>0.632, r is significant and the line may be usedfor prediction. If you view this example on a number line, it will help you.

Suppose you computed r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624<–0.532, r is significant and the line can be used for prediction

Suppose you computed r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811<0.776<0.811, r is not significant, and the line should not be used for prediction.

Third-exam vs final-exam example: critical value method

Consider the third exam/final exam example . The line of best fit is: ŷ = –173.51+4.83 x with r = 0.6631 and there are n = 11 data points. Can the regression line be used for prediction? Given a third-exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

• H ₀ : ρ = 0
• H _a : ρ ≠ 0
• α = 0.05

• Use the "95% Critical Value" table for r with df = n – 2 = 11 – 2 = 9.
• The critical values are –0.602 and +0.602
• Since 0.6631>0.602, r is significant.
• Decision: Reject the null hypothesis.
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.