Final answer:
In order to test for significance of a sample correlation coefficient at the 0.05 alpha level, one calculates a t statistic using the formula r√(n-2)/√(1-r²) and compares it to the critical value. A sample size of 25 with an r of 0.45 yields a p-value of 0.026, leading to the rejection of the null hypothesis, indicating a significant correlation.
Step-by-step explanation:
The question asks how to calculate the two-tailed critical value of r, which is the sample correlation coefficient, to test for significance. The two-tailed critical value is determined to assess whether the observed correlation could be due to chance. With a sample size of 25 and a correlation (r) of 0.45, one can use a t-test to determine significance at the alpha (α) level of 0.05. The formula to calculate the t statistic for the sample correlation coefficient is r√(n-2)/√(1-r²). Once the t statistic is calculated, you compare it against the critical value from the t distribution with n - 2 degrees of freedom. If the absolute value of the calculated t is greater than the critical t value, the result is considered significant and the null hypothesis of no correlation is rejected.
For the given problem, the computed t value is calculated as 0.45√(25-2)/√(1-0.45²). After finding the t value, you can use statistical tables or software to compare the p-value with the alpha level. Here, it's mentioned that the p-value is 0.026, which is less than α = 0.05, leading to the rejection of the null hypothesis. This indicates that there is sufficient evidence to conclude a significant linear relationship between the variables being studied.