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 .At the 5% significance level which of the following is the correct conclusion to the test? A) Reject H₀and conclude that the returns of these two mutual funds are correlated. B) Reject H₀and conclude that the returns of these two mutual funds are not correlated. C) Do not reject and conclude that the returns of these two mutual funds are not correlated. D) Do not reject and conclude that the returns of these two mutual funds are correlated.

Question

Quizplus · Accepted Answer

The null hypothesis is that there is no correlation between the returns of the Vanguard Energy Fund and the Vanguard Healthcare Fund, while the alternative hypothesis is that there is a correlation between the returns. 
The test statistic for testing the correlation coefficient is given by:
t = r * sqrt((n-2)/(1-r^2))
where r is the sample correlation coefficient between the two funds, and n is the number of observations (in this case, 5 years). 
The critical value for a two-tailed test with alpha = 0.05 and 3 degrees of freedom (n-2) is 3.182. 
Using the given data, we can calculate the sample correlation coefficient between the two funds as:
r = sS1U1B1xy / sqrt(sS1U1B1x * sS1U1B1y) = 447.68 / sqrt(35.77 * 13.34) = 0.778 
Substituting into the formula for the test statistic, we get:
t = 0.778 * sqrt(3/0.405) = 5.91 
Since the calculated value of t is greater than the critical value of 3.182, we reject the null hypothesis and conclude that there is a statistically significant correlation between the returns of the two mutual funds at the 5% significance level. Therefore, the correct answer is A).

The Following Data for Five Years of the Annual Returns