The following data for five years of the annual returns for two of Vanguard's mutual funds, the Vanguard Energy Fund (x) and the Vanguard Healthcare Fund (y), were given as s_x= 35.77, s_y= 13.34, s_xy= 447.68. The competing hypotheses are H₀: ρ_xy = 0, H_A: ρ_xy ≠ 0. At the 5% significance level which of the following is the correct conclusion to the test? A) Reject the null hypothesis; we can conclude that the returns of these two mutual funds are correlated. B) Reject the null hypothesis; we cannot conclude that the returns of these two mutual funds are not correlated. C) Do not reject the null hypothesis; we can conclude that the returns of these two mutual funds are correlated. D) Do not reject the null hypothesis; we cannot conclude that the returns of these two mutual funds are correlated

Question

Quizplus · Accepted Answer

The null hypothesis states that there is no correlation between the returns of the two mutual funds (ρ = 0). The alternative hypothesis states that there is a correlation between the returns of the two mutual funds (ρ ≠ 0).

The test is conducted at the 5% significance level, which means that the probability of rejecting the null hypothesis when it is true is 5%.

The test statistic to use is the sample correlation coefficient, which is calculated as follows:

r = sS1U1B1xy / sqrt(sS1U1B1x * sS1U1B1y)

Substituting the given values, we get:

r = 447.68 / sqrt(35.77 * 13.34) = 0.898

The critical value for a two-tailed test with n-2=3 degrees of freedom at a 5% significance level is approximately 3.182.

Since the calculated value of r (0.898) is greater than the critical value (3.182), we reject the null hypothesis.

Therefore, the correct conclusion is to reject the null hypothesis and conclude that the returns of these two mutual funds are correlated. Choice A is the correct answer.

The Following Data for Five Years of the Annual Returns