- A positive rr means that when xx increases, yy increases and when xx decreases, yy decreases (positive correlation).
- A negative rr means that when xx increases, yy decreases and when xx decreases, yy increases (negative correlation).
- An rr of zero means there is absolutely no linear relationship between xx and yy (no correlation).
- High correlation does not suggest that xx causes yy or yy causes xx. We say "correlation does not imply causation." For example, every person who learned
math in the 17th century is dead. However, learning math does not necessarily cause
death!
The 95% Critical Values of the Sample Correlation Coefficient Table at the end of
this chapter (before the Summary) may be used to give you a good idea of whether the
computed value of rr is significant or not. Compare rr to the appropriate critical value in
the table. If rr is significant, then you may want to use the line for prediction.
Suppose you computed r=0.801r=0.801 using n=10n=10 data points.
df=n-2=10 -2=8df=n-2=10 -2=8. The critical values associated with df=8df=8 are -0.632 and
+ 0.632. If rr<negative critical valuenegative critical value or r>positive critical valuer>positive critical value, then rr is
significant. Since r=0.801r=0.801 and 0.801>0.6320.801>0.632, rr is significant and the line may be used
for prediction. If you view this example on a number line, it will help you.
Suppose you computed r=-0.624r=-0.624 with 14 data points. df=14-2=12df=14-2=12. The critical values are -0.532 and 0.532. Since -0.624-0.624<-0.532-0.532, rr is significant and
the line may be used for prediction
Suppose you computed r=0.776r=0.776 and n=6n=6. df=6-2=4df=6-2=4. The
critical values are -0.811 and 0.811. Since -0.811-0.811< 0.7760.776 < 0.8110.811, rr is not significant
and the line should not be used for prediction.
If
rr is -1 or
rr is +1, then all the data points lie exactly on a straight line. If the line is significant, then
within the range of the x-values, the line can be used to predict a
yy value. As an illustration, consider the
third exam/final exam example.
The line of best fit is:
y
^
=
-173.51
+
4.83x
y
^
=-173.51+4.83x with
r
=
0.6631
r=0.6631
Can the line be used for prediction? Given a third exam score (xx value), can we
successfully predict the final exam score (predicted yy value). Test r=0.6631r=0.6631
with its appropriate critical value.
Using the table with df=11-2=9df=11-2=9, the critical values are -0.602 and +0.602. Since
0.6631>0.6020.6631>0.602, rr is significant. Because rr is significant and the scatter plot shows a reasonable linear trend, the line can be used to predict final exam scores.
Suppose you computed the following correlation coefficients. Using the
table at the end of the chapter, determine if rr is significant and the line of best fit associated
with each rr can be used to predict a yy value. If it helps, draw a number line.
- r=-0.567r=-0.567 and the sample size, nn, is 19. The df=n-2=17df=n-2=17. The critical value is -0.456. -0.567-0.567<-0.456-0.456 so rr is significant.
- r=0.708r=0.708 and the sample size, nn, is 9. The df=n-2=7df=n-2=7. The critical value is 0.666. 0.708>0.6660.708>0.666 so rr is significant.
- r=0.134r=0.134 and the sample size, nn, is 14. The df=14-2=12df=14-2=12. The critical value is 0.532. 0.134 is between -0.532 and 0.532 so rr is not significant.
- r=0r=0 and the sample size, nn, is 5. No matter what the dfs are, r=0r=0 is between the two critical values so rr is not significant.
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