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?

And the degrees of freedom are given by df=15-2=13

Step-by-step explanation:

Previous concepts

Pearson correlation coefficient(r), "measures a linear dependence between two variables (x and y). Its a parametric correlation test because it depends to the distribution of the data. And other assumption is that the variables x and y needs to follow a normal distribution".

The t distribution (Student’s t-distribution) is a "probability distribution that is used to estimate population parameters when the sample size is small (n<30) or when the population variance is unknown".

The shape of the t distribution is determined by its degrees of freedom and when the degrees of freedom increase the t distirbution becomes a normal distribution approximately.  

Solution to the problem

In order to calculate the correlation coefficient we can use this formula:

On this case we got that r =0.38

In order to test a hypothesis related to the correlation coeffcient we need to use the following statistic:

Where n represent the sample size and the statistic t follows a t distribution with n-2 degrees of freedom:

On this case our value of n = 15 and the statistic is given by:

And the degrees of freedom are given by df=15-2=13

c. 1,233 miles