The figure below shows that although the value of Pearson’s r has not changed, it is still very small (0.078), the relationship between the number of cups of tea drunk per day and cognitive function is now just significant (p = 0.038) if you use the common criterion of 𝛼=0.05, and the confidence intervals no longer cross zero (0.001, 0.152). (Although note that the lower confidence interval is very close to zero, suggesting that under the usual assumptions the effect in the population could be very close to zero.)
This example indicates one of the downfalls of significance testing; you can get significant results when you have large sample sizes even if the effect is very small. Basically, whether you get a significant result or not is at the mercy of the sample size.
| Variable | tea | cog_fun | |||||
|---|---|---|---|---|---|---|---|
| 1. tea | n | — | |||||
| Pearson's r | — | ||||||
| p-value | — | ||||||
| Lower 95% CI | — | ||||||
| Upper 95% CI | — | ||||||
| 2. cog_fun | n | 716 | — | ||||
| Pearson's r | 0.078 | * | — | ||||
| p-value | 0.038 | — | |||||
| Lower 95% CI | -8.628×10-4 | — | |||||
| Upper 95% CI | 0.152 | — | |||||
| Note. Confidence intervals based on 1000 bootstrap replicates. | |||||||
| * p < .05, ** p < .01, *** p < .001 | |||||||