Recently, an automobile insurance company performed a study of a random sample of 15 of its customers to determine if there is a positive relationship between the number of miles driven and the age of the driver. The sample correlation coefficient is r=.38. Given this information, and assuming that the test is to be performed at the .05 level of significance, which of the following is the correct test statistic?
Pearson r
1.771
To determine the correct test statistic for this situation, we need to perform a hypothesis test to assess the significance of the correlation coefficient (r) between the number of miles driven and the age of the driver.
The null hypothesis (H0) for this test is that there is no correlation between the two variables, meaning the population correlation coefficient (ρ) is 0. The alternative hypothesis (H1) is that there is a positive correlation between the two variables, meaning ρ is greater than 0.
Since the sample size is small (n = 15), the appropriate test statistic to use is the t-distribution with n-2 degrees of freedom. The formula for the test statistic (t) is:
t = (r * sqrt(n-2)) / sqrt(1 - r^2)
Given that the sample correlation coefficient is r = 0.38 and the sample size is n = 15, we can calculate the test statistic.
t = (0.38 * sqrt(15-2)) / sqrt(1 - 0.38^2)
t ≈ 1.75
Therefore, the correct test statistic is approximately 1.75.
To determine the correct test statistic, we need to clarify what hypothesis test is being conducted.
In this case, the study aims to determine if there is a positive relationship between the number of miles driven and the age of the driver. This suggests that we are interested in testing if there is a significant correlation between these two variables.
The appropriate hypothesis test for testing the significance of the correlation coefficient is the t-test. The test statistic associated with the t-test for correlation is called the t-value.
The formula to calculate the t-value for correlation is:
t = r * sqrt((n - 2) / (1 - r^2))
where r is the sample correlation coefficient and n is the sample size.
In this case, the given sample correlation coefficient is r = 0.38 and the sample size is n = 15.
Now we can substitute these values into the formula to calculate the t-value:
t = 0.38 * sqrt((15 - 2) / (1 - 0.38^2))
Calculating this expression, we get:
t ≈ 0.38 * sqrt(13 / (1 - 0.1444))
t ≈ 0.38 * sqrt(13 / 0.8556)
t ≈ 0.38 * sqrt(15.16)
t ≈ 0.38 * 3.889
t ≈ 1.477
So, the correct test statistic in this case is approximately t = 1.477.