Question

The following table shows a portion of the monthly returns data (in percent) for 2010–2016 for two of Vanguard’s mutual funds: the Vanguard Energy Fund and the Vanguard Healthcare Fund. [You may find it useful to reference the t table.]

Date	Energy	Healthcare
Jan-10	-4.89	-0.11
Feb-10	1.6	0.54
Mar-10	2.27	1.37
Apr-10	2.99	-3.84
May-10	-11.6	-5.16
Jun-10	-5.77	-0.52
Jul-10	8.69	1.52
Aug-10	-6.02	-1.08
Sep-10	10.2	8.24
Oct-10	3.85	2.3
Nov-10	2.82	-2.5
Dec-10	5.55	2
Jan-16	-1.44	-8.79
Feb-16	-2.57	-1.87
Mar-16	12.5	-0.46
Apr-16	10.03	2.62
May-16	-1.37	2.68
Jun-16	3.52	-0.03
Jul-16	-1.05	5.18
Aug-16	2.51	-4.88
Sep-16	2.6	0.56
Oct-16	-3.05	-7.63
Nov-16	7.01	1.51
Dec-16	0.2	-5.29

a. Calculate the sample correlation coefficient r_xy. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
b. Specify the competing hypotheses in order to determine whether the population correlation coefficient is different from zero.

• H₀: ρ_xy ≤ 0; H_A: ρ_xy > 0

• H₀: ρ_xy ≥ 0; H_A: ρ_xy < 0

• H₀: ρ_xy = 0; H_A: ρ_xy ≠ 0

c-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

c-2. Find the p-value.

• p-value < 0.01

• 0.01 ≤ p-value < 0.02

• 0.02 ≤ p-value < 0.05

• 0.05 ≤ p-value < 0.10

• p-value ≥ 0.10

c-3. At the 5% significance level, what is the conclusion to the test?

• Reject H₀; there is enough evidence to state the returns are correlated.

• Reject H₀; there is not enough evidence to state the returns are correlated.

• Do not reject H₀; there is enough evidence to state the returns are correlated.

• Do not reject H₀; there is not enough evidence to state the returns are correlated.

Answer 1