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.

a. Calculate the sample correlation coefficient r_xy.

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.

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.

Date   Energy   Healthcare
Jan-10   -4.94   -0.18
Feb-10   1.7   0.38
Mar-10   2.45   1.21
Apr-10   3.12   -3.84
May-10   -11   -5.12
Jun-10   -5.91   -0.61
Jul-10   8.87   1.44
Aug-10   -5.82   -1.11
Sep-10   10.13   8.32
Oct-10   3.99   2.37
Nov-10   2.79   -2.45
Dec-10   5.68   0.9
Jan-11   6.45   1.75
Feb-11   5.9   2.98
Mar-11   1.42   1.19
Apr-11   1.69   5.72
May-11   -4.17   2.36
Jun-11   -1.97   -0.59
Jul-11   1.15   -2.74
Aug-11   -10.42   -2.57
Sep-11   -14.94   -3.93
Oct-11   18.71   4.6
Nov-11   0.87   -0.16
Dec-11   -8.45   -3.06
Jan-12   4.3   2.38
Feb-12   4.92   1.57
Mar-12   -6.56   3.61
Apr-12   -1.68   -0.43
May-12   -12.16   -3.47
Jun-12   5.48   5.47
Jul-12   2.79   -0.33
Aug-12   2.79   1.85
Sep-12   2.94   3.95
Oct-12   -0.91   -0.71
Nov-12   -2.5   0.45
Dec-12   2.4   -3.26
Jan-13   5.7   6.45
Feb-13   -2.05   1.27
Mar-13   1.58   3.64
Apr-13   -0.18   3.01
May-13   1.71   1.42
Jun-13   -3.49   0.2
Jul-13   5.32   5.59
Aug-13   -0.22   -1.93
Sep-13   2.95   4.14
Oct-13   4.59   4
Nov-13   -0.92   4.8
Dec-13   -1.71   -4.74
Jan-14   -5.07   2.23
Feb-14   5.83   8.49
Mar-14   1.84   -6.03
Apr-14   5.41   -3.2
May-14   1.32   3.7
Jun-14   5.5   3.31
Jul-14   -4.68   0.04
Aug-14   1.69   3.98
Sep-14   -7.23   -0.21
Oct-14   -4.98   5.44
Nov-14   -9.07   3.44
Dec-14   -9.68   -8.41
Jan-15   -4.09   1.94
Feb-15   5.01   4.84
Mar-15   -2.58   -0.35
Apr-15   10.56   -0.69
May-15   -6.62   4.66
Jun-15   -4.19   -0.21
Jul-15   -7.58   2.65
Aug-15   -4.83   -5.75
Sep-15   -7.66   -5.82
Oct-15   10.8   5.83
Nov-15   -1   1.52
Dec-15   -11.57   -3.87
Jan-16   -1.2   -8.75
Feb-16   -2.48   -1.85
Mar-16   12.61   -0.4
Apr-16   10.23   2.59
May-16   -1.51   2.49
Jun-16   3.58   -0.14
Jul-16   -1.13   5.27
Aug-16   2.46   -4.94
Sep-16   2.39   0.58
Oct-16   -3.03   -7.53
Nov-16   6.92   1.44
Dec-16   -0.5   -5.14

Answer 1

(a) the sample correlation coefficient r_xy=0.4285

(b)H₀: ρ_xy = 0; H_A: ρ_xy ≠ 0

(c1) the test statistic t=4.294

we use t-test here and

t =r/sqrt[(1—r²)/(n—2)]=0.4285/sqrt((1-0.4285*0.4285)/(84-2))=4.294 with n-2=84-2=82 df

(c2)p-value < 0.01

p-value=0.000048 ( using ms-excel=tdist(4.295,82,2))

(c3) Reject H₀; there is enough evidence to state the returns are correlated.

since the p-value is less than alpha=0.05

correlation coefficient is calculated using ms-excel function

	Energy(x)	Healthcare(y)
n=	84	84
correlation(x,y)=r=	0.4285