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In: Statistics and Probability

The following table shows a portion of the monthly returns data (in percent) for 2010–2016 for...

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 −0.08 Feb-10 1.90 0.56 ⋮ ⋮ ⋮ Dec-16 0.10 −5.30 SOURCE: www.finance.yahoo.com. Click here for the Excel Data File a. Calculate the sample correlation coefficient rxy. (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: ρxy ≤ 0; HA: ρxy > 0 H0: ρxy = 0; HA: ρxy ≠ 0 H0: ρxy ≥ 0; HA: ρxy < 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.

Date Energy Healthcare
Jan-10 -4.86 -0.08
Feb-10 1.9 0.56
Mar-10 2.3 1.41
Apr-10 3.02 -3.76
May-10 -11.2 -5.13
Jun-10 -5.77 -0.51
Jul-10 8.75 1.55
Aug-10 -5.94 -0.99
Sep-10 10.14 8.29
Oct-10 3.87 2.33
Nov-10 2.82 -2.49
Dec-10 5.56 2.3
Jan-11 6.47 1.75
Feb-11 6.6 2.97
Mar-11 1.41 1.32
Apr-11 1.63 5.81
May-11 -4.1 2.62
Jun-11 -2.02 -0.61
Jul-11 1.14 -2.66
Aug-11 -10.4 -2.77
Sep-11 -15.04 -4.02
Oct-11 18.71 4.6
Nov-11 0.95 -0.12
Dec-11 -8.41 -3
Jan-12 4.4 2.45
Feb-12 5.08 1.67
Mar-12 -6.54 3.44
Apr-12 -1.59 -0.47
May-12 -12.21 -3.54
Jun-12 5.65 5.46
Jul-12 2.81 -0.46
Aug-12 2.86 1.81
Sep-12 2.83 3.86
Oct-12 -1.08 -0.77
Nov-12 -1.3 0.5
Dec-12 0.2 -3.02
Jan-13 5.68 6.52
Feb-13 -2.05 1.37
Mar-13 1.55 3.57
Apr-13 -0.13 3.01
May-13 1.64 1.35
Jun-13 -3.64 0.34
Jul-13 5.26 5.6
Aug-13 -0.27 -2.07
Sep-13 3.01 4.1
Oct-13 4.62 4
Nov-13 -0.83 6.1
Dec-13 -1.68 -4.69
Jan-14 -5.2 2.4
Feb-14 6.05 8.58
Mar-14 1.63 -6.09
Apr-14 5.5 -2.5
May-14 1.13 3.57
Jun-14 5.2 3.44
Jul-14 -4.72 -0.01
Aug-14 1.77 4.05
Sep-14 -7.46 -0.19
Oct-14 -5.03 5.24
Nov-14 -9.11 3.38
Dec-14 -9.7 -8.45
Jan-15 -3.9 2.03
Feb-15 4.94 4.8
Mar-15 -2.54 -0.42
Apr-15 10.6 -0.7
May-15 -6.54 4.7
Jun-15 -4.31 -0.25
Jul-15 -7.68 2.45
Aug-15 -4.81 -5.8
Sep-15 -7.67 -5.76
Oct-15 10.81 5.97
Nov-15 -0.97 1.45
Dec-15 -11.69 -3.71
Jan-16 -1.44 -8.82
Feb-16 -2.55 -1.82
Mar-16 12.51 -0.41
Apr-16 10.11 2.58
May-16 -1.42 2.59
Jun-16 3.56 -0.05
Jul-16 -1.02 5.18
Aug-16 2.57 -4.88
Sep-16 2.51 0.64
Oct-16 -2.99 -7.58
Nov-16 7.03 1.51
Dec-16 0.1 -5.3

Solutions

Expert Solution

Following table shows the calculations:

Date Energy, X Healthcare, Y X^2 Y^2 XY
Jan-10 -4.86 -0.08 23.6196 0.0064 0.3888
Feb-10 1.9 0.56 3.61 0.3136 1.064
Mar-10 2.3 1.41 5.29 1.9881 3.243
Apr-10 3.02 -3.76 9.1204 14.1376 -11.3552
May-10 -11.2 -5.13 125.44 26.3169 57.456
Jun-10 -5.77 -0.51 33.2929 0.2601 2.9427
Jul-10 8.75 1.55 76.5625 2.4025 13.5625
Aug-10 -5.94 -0.99 35.2836 0.9801 5.8806
Sep-10 10.14 8.29 102.8196 68.7241 84.0606
Oct-10 3.87 2.33 14.9769 5.4289 9.0171
Nov-10 2.82 -2.49 7.9524 6.2001 -7.0218
Dec-10 5.56 2.3 30.9136 5.29 12.788
Jan-11 6.47 1.75 41.8609 3.0625 11.3225
Feb-11 6.6 2.97 43.56 8.8209 19.602
Mar-11 1.41 1.32 1.9881 1.7424 1.8612
Apr-11 1.63 5.81 2.6569 33.7561 9.4703
May-11 -4.1 2.62 16.81 6.8644 -10.742
Jun-11 -2.02 -0.61 4.0804 0.3721 1.2322
Jul-11 1.14 -2.66 1.2996 7.0756 -3.0324
Aug-11 -10.4 -2.77 108.16 7.6729 28.808
Sep-11 -15.04 -4.02 226.2016 16.1604 60.4608
Oct-11 18.71 4.6 350.0641 21.16 86.066
Nov-11 0.95 -0.12 0.9025 0.0144 -0.114
Dec-11 -8.41 -3 70.7281 9 25.23
Jan-12 4.4 2.45 19.36 6.0025 10.78
Feb-12 5.08 1.67 25.8064 2.7889 8.4836
Mar-12 -6.54 3.44 42.7716 11.8336 -22.4976
Apr-12 -1.59 -0.47 2.5281 0.2209 0.7473
May-12 -12.21 -3.54 149.0841 12.5316 43.2234
Jun-12 5.65 5.46 31.9225 29.8116 30.849
Jul-12 2.81 -0.46 7.8961 0.2116 -1.2926
Aug-12 2.86 1.81 8.1796 3.2761 5.1766
Sep-12 2.83 3.86 8.0089 14.8996 10.9238
Oct-12 -1.08 -0.77 1.1664 0.5929 0.8316
Nov-12 -1.3 0.5 1.69 0.25 -0.65
Dec-12 0.2 -3.02 0.04 9.1204 -0.604
Jan-13 5.68 6.52 32.2624 42.5104 37.0336
Feb-13 -2.05 1.37 4.2025 1.8769 -2.8085
Mar-13 1.55 3.57 2.4025 12.7449 5.5335
Apr-13 -0.13 3.01 0.0169 9.0601 -0.3913
May-13 1.64 1.35 2.6896 1.8225 2.214
Jun-13 -3.64 0.34 13.2496 0.1156 -1.2376
Jul-13 5.26 5.6 27.6676 31.36 29.456
Aug-13 -0.27 -2.07 0.0729 4.2849 0.5589
Sep-13 3.01 4.1 9.0601 16.81 12.341
Oct-13 4.62 4 21.3444 16 18.48
Nov-13 -0.83 6.1 0.6889 37.21 -5.063
Dec-13 -1.68 -4.69 2.8224 21.9961 7.8792
Jan-14 -5.2 2.4 27.04 5.76 -12.48
Feb-14 6.05 8.58 36.6025 73.6164 51.909
Mar-14 1.63 -6.09 2.6569 37.0881 -9.9267
Apr-14 5.5 -2.5 30.25 6.25 -13.75
May-14 1.13 3.57 1.2769 12.7449 4.0341
Jun-14 5.2 3.44 27.04 11.8336 17.888
Jul-14 -4.72 -0.01 22.2784 0.0001 0.0472
Aug-14 1.77 4.05 3.1329 16.4025 7.1685
Sep-14 -7.46 -0.19 55.6516 0.0361 1.4174
Oct-14 -5.03 5.24 25.3009 27.4576 -26.3572
Nov-14 -9.11 3.38 82.9921 11.4244 -30.7918
Dec-14 -9.7 -8.45 94.09 71.4025 81.965
Jan-15 -3.9 2.03 15.21 4.1209 -7.917
Feb-15 4.94 4.8 24.4036 23.04 23.712
Mar-15 -2.54 -0.42 6.4516 0.1764 1.0668
Apr-15 10.6 -0.7 112.36 0.49 -7.42
May-15 -6.54 4.7 42.7716 22.09 -30.738
Jun-15 -4.31 -0.25 18.5761 0.0625 1.0775
Jul-15 -7.68 2.45 58.9824 6.0025 -18.816
Aug-15 -4.81 -5.8 23.1361 33.64 27.898
Sep-15 -7.67 -5.76 58.8289 33.1776 44.1792
Oct-15 10.81 5.97 116.8561 35.6409 64.5357
Nov-15 -0.97 1.45 0.9409 2.1025 -1.4065
Dec-15 -11.69 -3.71 136.6561 13.7641 43.3699
Jan-16 -1.44 -8.82 2.0736 77.7924 12.7008
Feb-16 -2.55 -1.82 6.5025 3.3124 4.641
Mar-16 12.51 -0.41 156.5001 0.1681 -5.1291
Apr-16 10.11 2.58 102.2121 6.6564 26.0838
May-16 -1.42 2.59 2.0164 6.7081 -3.6778
Jun-16 3.56 -0.05 12.6736 0.0025 -0.178
Jul-16 -1.02 5.18 1.0404 26.8324 -5.2836
Aug-16 2.57 -4.88 6.6049 23.8144 -12.5416
Sep-16 2.51 0.64 6.3001 0.4096 1.6064
Oct-16 -2.99 -7.58 8.9401 57.4564 22.6642
Nov-16 7.03 1.51 49.4209 2.2801 10.6153
Dec-16 0.1 -5.3 0.01 28.09 -0.53
Total 7.07 51.32 3129.908 1216.928 853.7943

Sample size: n=84

Now,

The coefficient of correlation is :

(b)

Hypotheses are:

H0: ρxy = 0; HA: ρxy ≠ 0

(c)

Test statistics:

The p-value is

p-value < 0.01

c-3:

Since p-value is less than 0.05 so we reject the null hypothesis.

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

orchestra answered 2 years ago
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