In: Statistics and Probability
The following 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, -4.95, -0.19); (Feb-10, 1.8, 0.39); (Mar-10, 2.47, 1.18); (Apr-10, 3.14, -3.8); (May-10, -11.5, -5.09); (Jun-10, -5.95, -0.63); (Jul-10, 8.83, 1.47); (Aug-10, -5.84, -1.1); (Sep-10, 10.13, 8.32); (Oct-10, 3.98, 2.37); (Nov-10, 2.81, -2.41); (Dec-10, 5.71, 1.00); (Jan-11, 6.42, 1.74); (Feb-11, 5.7, 2.99); (Mar-11, 1.37, 1.17); (Apr-11, 1.73, 5.69); (May-11, -4.22, 2.41); (Jun-11, -1.98, -0.59); (Jul-11, 1.19, -2.75); (Aug-11, -10.42, -2.61); (Sep-11, -14.9, -3.89); (Oct-11, 18.74, 4.64); (Nov-11, 0.82, -0.21); (Dec-11, -8.49, -3.08); (Jan-12, 4.28, 2.42); (Feb-12, 4.91, 1.56); (Mar-12, -6.51, 3.62); (Apr-12, -1.72, -0.47); (May-12, -12.15, -3.48); (Jun-12, 5.44, 5.45); (Jul-12, 2.8, 0.31); (Aug-12, 2.83, 1.88); (Sep-12, 2.98, 3.93); (Oct-12, -0.96, -0.73); (Nov-12, -2.8, 0.41); (Dec-12, 2.2, -3.28); (Jan-13, 5.74, 6.47); (Feb-13, -2.02, 1.3); (Mar-13, 1.59, 3.6); (Apr-13, -0.15, 2.99); (May-13, 1.69, 1.42); (Jun-13, -3.49, 0.22); (Jul-13, 5.31, 5.54); (Aug-13, -0.21, -1.92); (Sep-13, 2.95, 4.1); (Oct-13, 4.57, 3.97); (Nov-13, -0.82, 5.2); (Dec-13, -1.74, -4.7); (Jan-14, -5.1, 2.25); (Feb-14, 5.86, 8.54); (Mar-14, 1.88, -5.98); (Apr-14, 5.46, -3.1); (May-14, 1.3, 3.73); (Jun-14, 5.5, 3.35); (Jul-14, -4.72, 0.04); (Aug-14, 1.7, 3.94); (Sep-14, -7.23, -0.19); (Oct-14, -4.95, 5.45); (Nov-14, -9.09, 3.4); (Dec-14, -9.73, -8.36); (Jan-15, -4.11, 1.95); (Feb-15, 5.05, 4.8); (Mar-15, -2.6, -0.35); (Apr-15, 10.52, -0.73); (May-15, -6.62, 4.71); (Jun-15, -4.17, -0.26); (Jul-15, -7.59, 2.64); (Aug-15, -4.87, -5.79); (Sep-15, -7.64, -5.8); (Oct-15, 10.79, 5.87); (Nov-15, -1.03, 1.47); (Dec-15, -11.59, -3.91); (Jan-16, -1.2, -8.79); (Feb-16, -2.52, -1.84); (Mar-16, 12.63, -0.42); (Apr-16, 12.63, 2.56); (May-16, -1.55, 2.52); (Jun-16, 3.62, -0.11); (Jul-16, -1.08, 5.28); (Aug-16, 2.42, -4.97); (Sep-16, 2.35, 0.54); (Oct-16, -3.06, -7.51); (Nov-16, 6.9, 1.48); (Dec-16, -0.8, -5.15)
a. Calculate the sample correlation coefficient Rxy. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
Sample Correlation Coefficient = _________
b. Calculate the value of the Test Statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
Test Statistic = _________
(I can attach the spreadsheet in a comment. It made it too long to post as one question.)
The data has been put in an excel spreadsheet and then computed using the standard formulae. Here, since sample correlation coefficient is required, I've used the formula STDEV which uses (n-1) as the denominator. For finding the covariance between the two series of values, the function used is COVAR which is basically the standard statistical formula using 'n' in the denominator. Then Rxy= Cov(X,Y)/(SD(X)*SD(Y)) is computed.
The test statistic for two series of data being correlated is given by r(sqrt(n-2))/(sqrt(1-r*r)), where correlation ratio Rxy is given by r and n denotes the number of observations.
The spreadsheet is added as a picture.
The test statistic is equal to 4.28 and the value of t(0.025,82)=1.99 approx. Here, the level of significance is taken to be 5% and degrees of freedom is 82. Also, this is a both tailed test. Since, modulus of the value of test statistic is greater than the tabulated t value, we reject the null hypothesis of Rxy=0 and conclude at 5% level of significance that there is significant correlation between the two series of data.