Question

In a sample of 300 steel rods, the correlation coefficient between diameter and length was r = 0.15. Find the P-value for testing H0 : ρ ≤ 0 vs. H1 : ρ > 0. Can we conclude that ρ > 0? (Round the final answer to three decimal places.)

Answer 1

The p-value for the correlation coefficient between diameter and length in a sample of steel rods is 0.026, which is lower than the significance level of 0.05, leading us to reject the null hypothesis and conclude that there is a significant positive correlation between these variables.

Step-by-step explanation:

The correlation coefficient between diameter and length in a sample of 300 steel rods is given as r = 0.15. To test the null hypothesis H0: ρ ≤ 0 against the alternative hypothesis H1: ρ > 0, we can use a t-test for the significance of the population correlation coefficient.

The given p-value is 0.026, which is less than the typical significance level of α = 0.05. This leads us to reject the null hypothesis (H0), suggesting there is sufficient evidence to conclude that there is a significant linear relationship between diameter and length of steel rods. Therefore, we can conclude that ρ (rho), the population correlation coefficient, is greater than zero.

To compare the sample correlation coefficient r with critical values, degrees of freedom, df, would be n - 2 = 300 - 2 = 298. In this case, we directly have the p-value and thus do not need to refer to a table of critical values. Since the p-value is significant, we can say that diameter can be used to predict length in this case.