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	-4.86	-0.13
Feb-10	1.5	0.58
Mar-10	2.29	1.44
Apr-10	2.98	-3.71
May-10	-11.4	-5.15
Jun-10	-5.56	-0.42
Jul-10	8.74	1.55
Aug-10	-6.08	-0.96
Sep-10	10.11	8.16
Oct-10	3.91	2.18
Nov-10	2.88	-2.53
Dec-10	5.51	1.4
Jan-11	6.64	1.55
Feb-11	5.9	3.02
Mar-11	1.34	1.17
Apr-11	1.56	5.94
May-11	-4.02	2.53
Jun-11	-2.11	-0.56
Jul-11	1.17	-2.69
Aug-11	-10.36	-2.67
Sep-11	-15.03	-4.22
Oct-11	18.64	4.58
Nov-11	1.03	-0.08
Dec-11	-8.37	-2.95
Jan-12	4.39	2.51
Feb-12	5.16	1.59
Mar-12	-6.49	3.52
Apr-12	-1.45	-0.47
May-12	-12.23	-3.47
Jun-12	5.75	5.52
Jul-12	2.68	-0.41
Aug-12	2.85	1.96
Sep-12	2.96	3.68
Oct-12	-1.11	-0.69
Nov-12	-1.7	0.37
Dec-12	-0.5	-2.97
Jan-13	5.81	6.51
Feb-13	-1.88	1.27
Mar-13	1.58	3.53
Apr-13	-0.26	3.23
May-13	1.56	1.25
Jun-13	-3.57	0.44
Jul-13	5.38	5.63
Aug-13	-0.36	-2.09
Sep-13	3.04	3.99
Oct-13	4.48	3.84
Nov-13	-0.91	4.6
Dec-13	-1.46	-4.61
Jan-14	-5.17	2.39
Feb-14	6.11	8.46
Mar-14	1.52	-6.07
Apr-14	5.41	-1.9
May-14	1.31	3.69
Jun-14	4.3	3.4
Jul-14	-4.73	-0.17
Aug-14	1.82	4.13
Sep-14	-7.39	-0.33
Oct-14	-5.07	5.17
Nov-14	-9.01	3.45
Dec-14	-9.76	-8.53
Jan-15	-3.95	2.09
Feb-15	5.12	4.81
Mar-15	-2.53	-0.37
Apr-15	10.44	-0.54
May-15	-6.35	4.76
Jun-15	-4.28	-0.37
Jul-15	-7.79	2.57
Aug-15	-4.85	-5.81
Sep-15	-7.52	-5.71
Oct-15	10.78	5.83
Nov-15	-0.85	1.37
Dec-15	-11.83	-3.76
Jan-16	-1.53	-8.93
Feb-16	-2.52	-1.98
Mar-16	12.33	-0.38
Apr-16	10.05	2.64
May-16	-1.37	2.74
Jun-16	3.54	-0.03
Jul-16	-1.07	5.18
Aug-16	2.54	-4.9
Sep-16	2.67	0.68
Oct-16	-2.97	-7.66
Nov-16	7.04	1.51
Dec-16	-0.3	-5.26

H₀: ρ_xy = 0; H_A: ρ_xy ≠ 0b. 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

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

Answer 1

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

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

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.)

Excel used.

Correlation Matrix
		Energy	Healthcare
	Energy	1.000
	Healthcare	0.4489	1.000
		84	sample size

t=4.5491

test statistic t = 4.55 ( two decimals)

DF = n-2 =82

Table value of t with 82 DF at 0.01 level = 2.637

Calculated t =4.55 < 2.637 , the table value

The null hypothesis is rejected at 0.01 level.

c-2. Find the p-value.

Answer: 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