A data set of size 21 has correlation coefficient of r = 0.476. Test the significance of r at the 5%
Level and at the 1% level.
A)r is significant at 5%, but not at 1%.
B)r is not significant at 5% or at 1%.
C)r is significant at 5% and at 1%.

Question

Quizplus · Accepted Answer

To test the significance of the correlation coefficient at the 5% and 1% levels, we need to perform a hypothesis test. Null hypothesis: The population correlation coefficient is zero (i.e., there is no significant correlation). Alternative hypothesis: The population correlation coefficient is non-zero (i.e., there is a significant correlation). Using a two-tailed test, we can use the t-distribution with n-2 degrees of freedom to obtain the critical values. For a 5% significance level, the critical value is ±2.085. For a 1% significance level, the critical value is ±2.831. Next, we calculate the t-statistic for the sample correlation coefficient using the formula: t = r * sqrt(n-2) / sqrt(1-r^2) Plugging in the values, we get: t = 0.476 * sqrt(21-2) / sqrt(1-0.476^2) = 2.189 Since t > 2.085, we can reject the null hypothesis at the 5% level and conclude that there is a significant correlation. However, since t < 2.831, we cannot reject the null hypothesis at the 1% level. Therefore, the answer is A, r is significant at 5%, but not at 1%. To test the significance of r, we use the t-test for correlation coefficient. For n=21, r=0.476, and a two-tailed test at the 5% level, the critical t-value is ±2.085 (from a t-table or calculator). The calculated t-value is t=2.168 (using the formula t=r√(n-2)/√(1-r^2)), which is greater than the critical t-value. Since 2.168 > 2.085, we reject the null hypothesis and conclude that r is significant at the 5% level. For a two-tailed test at the 1% level, the critical t-value is ±2.831. Since 2.168 < 2.831, we fail to reject the null hypothesis and conclude that r is not significant at the 1% level.

A Data Set of Size 21 Has Correlation Coefficient of R