Step 1 :We are given the ages and bone densities of 5 random women at a hospital. The ages are [33, 37, 45, 53, 61] and the corresponding bone densities are [345, 335, 325, 320, 315].
Step 2 :We need to determine if the correlation coefficient is statistically significant at the 0.05 level. To do this, we first calculate the correlation coefficient.
Step 3 :The correlation coefficient, denoted as r, is calculated to be -0.9642745893155453.
Step 4 :We then compare this with the critical value for a two-tailed test at the 0.05 level, which is approximately ±0.878 for a sample size of 5.
Step 5 :If the absolute value of the correlation coefficient is greater than the critical value, then it is statistically significant.
Step 6 :The absolute value of our calculated correlation coefficient is greater than the critical value (|-0.964| > 0.878).
Step 7 :\(\boxed{\text{Therefore, the correlation coefficient is statistically significant at the 0.05 level.}}\)