Use the table of critical values of the Pearson correlation coefficient to determine whether the correlation coefficient is statistically significant at the specified level of significance for the given sample size. Write the critical value to the nearest thousandth. \[ r=0.612, \alpha=0.05, n=10 \]

Step 1 ：The Pearson correlation coefficient, r, measures the strength and direction of the linear relationship between two variables. The value of r is always between -1 and 1. In this case, we are given r=0.612, which indicates a moderate positive correlation.

Step 2 ：The level of significance, α, is the probability of rejecting the null hypothesis when it is true. In this case, α=0.05, which means there is a 5% chance of rejecting the null hypothesis when it is true.

Step 3 ：The sample size, n, is the number of observations in our sample. In this case, n=10.

Step 4 ：To determine whether the correlation coefficient is statistically significant, we need to compare the absolute value of r to the critical value from the table of critical values of the Pearson correlation coefficient. If the absolute value of r is greater than the critical value, then the correlation is statistically significant.

Step 5 ：The critical value for a two-tailed test with α=0.05 and n=10 is approximately 0.632.

Step 6 ：The absolute value of the correlation coefficient, 0.612, is less than the critical value, 0.632. Therefore, the correlation is not statistically significant at the 0.05 level of significance for a sample size of 10.

Step 7 ：Final Answer: The correlation is not statistically significant at the 0.05 level of significance for a sample size of 10. Therefore, the critical value to the nearest thousandth is \(\boxed{0.632}\).