For relationships between quantitative variables, where Pearson’s r is used to describe the strength of those relationships, the appropriate null hypothesis test is a test of Pearson’s r. The basic logic is exactly the same as for other null hypothesis tests. In this case, the null hypothesis is that there is no relationship in the population. We can use the Greek lowercase rho (ρ) to represent the relevant parameter: ρ = 0. The alternative hypothesis is that there is a relationship in the population: ρ ≠ 0. As with the t test, this test can be two-tailed if the researcher has no expectation about the direction of the relationship or one-tailed if the researcher expects the relationship to go in a particular direction.
It is possible to use Pearson’s r for the sample to compute a t score with N− 2 degrees of freedom and then to proceed as for a t test. However, because of the way it is computed, Pearson’s r can also be treated as its own test statistic. The online statistical tools and statistical software such as Excel and SPSS generally compute Pearson’s r and provide the p value associated with that value of Pearson’s r. As always, if the p value is less than .05, we reject the null hypothesis and conclude that there is a relationship between the variables in the population. If the p value is greater than .05, we retain the null hypothesis and conclude that there is not enough evidence to say there is a relationship in the population. If we compute Pearson’s rby hand, we can use a table like Table 13.5, which shows the critical values of rfor various samples sizes when α is .05. A sample value of Pearson’s rthat is more extreme than the critical value is statistically significant.
Critical value of r |
||
N |
One-tailed |
Two-tailed |
5 |
.805 |
.878 |
10 |
.549 |
.632 |
15 |
.441 |
.514 |
20 |
.378 |
.444 |
25 |
.337 |
.396 |
30 |
.306 |
.361 |
35 |
.283 |
.334 |
40 |
.264 |
.312 |
45 |
.248 |
.294 |
50 |
.235 |
.279 |
55 |
.224 |
.266 |
60 |
.214 |
.254 |
65 |
.206 |
.244 |
70 |
.198 |
.235 |
75 |
.191 |
.227 |
80 |
.185 |
.220 |
85 |
.180 |
.213 |
90 |
.174 |
.207 |
95 |
.170 |
.202 |
100 |
.165 |
.197 |
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