Question in Statistics...thanks!?

Question

Suppose 18 data (x, y) yielded a sample correlation coefficient of r = .50. Test the hypothesis that there is a significant linear relationship between x and y at the a = .01 significance level.

a. Write the appropriate null and alternative hypotheses.

b. Show the critical value of the test statistic and write the rejection rule for the null hypothesis.

c. Calculate the appropriate test statistic.

d. Write a conclusion about the relationship between x and y.

Answer 1

Hypothesis Test for correlation

assuming you have a large enough sample such that the central limit theorem holds and the mean is normally distributed then to test the null hypothesis H0: ρ = 0

find the test statistic z = r / sqrt((1 - r²)/(n-1))

where r is the regression coefficient

n is the sample size

The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.

H1: ρ ≠ 0; p-value is the area in the tails greater than |z|

for a small sample test for the mean every thing is the same save the test statistic is a t statistic with n - 2 degrees of freedom.  

the test statistic is

t = 0.50 / sqrt((1 - 0.5^2) / (18 - 1)) = 2.380476

the p-value of the test is:

2 * P(t_16 < -2.380476) = 0.03006377

because the p-value is greater than the significance level we fail to reject the null hypothesis and conclude that it is plausible there is no linear relationship.