Question

If the hypothesis that a correlation coefficient is equal to zero cannot be rejected with a sample size of 10, would the hypothesis be more or less likely to be rejected with a sample size of 100 and the same sample correlation value?

More likely

Less likely

Sample size does not impact the test

Not enough information to make a conclusion

Answer 1

Here curve with black dots indicates t-distribution with 8 df and curve with red line indicates t-distribution with 98 df. If the null hypothesis is accepted when sample size 10, then observed t lies inside the region bounded by curve with black dots and (-2.306, 2.306) where critical value at 0.05 level of significance and sample size 10=-2.306 or 2.306 (i.e. we reject null hypothesis if observed t<-2.306 or >2.306 i.e. observed t lies outside the region mentioned above). Now look at the curve with red line. We observe that probability of accepting null hypothesis at n=100 is larger than the probability of accepting null hypothesis at n=10. Hence If the hypothesis that a correlation coefficient is equal to zero cannot be rejected with a sample size of 10, the hypothesis be less likely to be rejected with a sample size of 100 and the same sample correlation value.