It is not obvious from the formulas, but the values of the sample test statistic t for the correlation coefficient and for the slope of the least-squares line are equal for the same data set. This fact is based on the relation

b = r sy/sx

where sy and sx are the sample standard deviations of the x and y values, respectively.

(a) Many computer software packages give the t value and corresponding P-value for b. If b is significant, is r significant?

(b) When doing statistical tests "by hand," it is easier to compute the sample test statistic t for the sample correlation coefficient r than it is to compute the sample test statistic t for the slope b of the sample least-squares line. Compare the result of parts (b) and (f) for problems 7-12 of this problem set. Is the sample test statistic t for r the same as the corresponding test statistic for b? If you conclude that r is positive, can you conclude that b is positive at the same level of significance? If you conclude that r is not significant, is b also not significant at the same level of significance?