Question

Column Row 159.26 463.54 161.02 483.55 124.84575.85 147.71567.66 193.22 567.81 306.68 616.68 612.56 720.15 460.03 782.62 375.67 728.84 424.35 979.52 360.48 977.33 103.9 49.3 53.8 56.9 60.6 65.2 72.6 82.4 90.9 96.5 99.6 2 4 9 10

Answer 1

Solution:

8) We have to find covariance between Xi,1 and Xi,2. That is, covariance between the first two columns.

Cov((Xi,1),(Xi,2)) = E((Xi,1)*(Xi,2)) - E(Xi,1)*E(Xi,2)

Where E(Xi,j) is expectation of Xi,j. Also, notice that number of observations for this case, n = 11. (Since number of rows is 11).

From the given table then, we form another table with three columns: Xi,1, Xi,2, and product of Xi,1 and Xi,2:

Xi,1	Xi,2	Xi,1*Xi,2
159.26	463.54	159.26*463.54 = 73823.38
161.02	483.55	161.02*483.55 = 77861.22
124.84	575.85	124.84*575.85 = 71889.11
147.71	567.66	147.71*567.66 = 83849.06
193.22	567.81	193.22*567.81 = 109712.25
306.68	616.68	306.68*616.68 = 189123.42
612.56	720.15	612.56*720.15 = 441135.08
460.03	782.62	460.03*782.62 = 360028.68
375.67	728.84	375.67*728.84 = 273803.32
424.35	979.52	424.35*979.52 = 415659.31
360.48	977.33	360.48*977.33 = 352307.92
Sum = 3325.82	Sum = 7463.55	Sum = 2449192.75

So, E(Xi,1) = sum(Xi,1)/n = 3325.82/11 = 302.35

And E(Xi,2) = sum(Xi,2)/n = 7463.55/11 = 678.50

And finally, E((Xi,1)(Xi,2)) = sum(Xi,1*Xi,2)/n = 2449192.75/11 = 222653.89

So, using the above mentioned formula:

covariance = (222653.89) - (302.35*678.50)

Cov = 222653.89 - 205144.48 = 17509.41

9) Correlation Coefficient between the two

r = Cov(Xi,1, Xi,2)/(variance(Xi,1)*variance(Xi,2))^1/2

Finding the required variances then:

Variance(Xi,j) = E(Xi,j²) - (E(Xi,j))²

We form another table carrying (Xi,1)² and (Xi,2)²:

Xi,1	Xi,12	Xi,2	Xi,22
159.26	25363.75	463.54	214869.33
161.02	25927.44	483.55	233820.60
124.84	15585.03	575.85	331603.22
147.71	21818.24	567.66	322237.88
193.22	37333.97	567.81	322408.20
306.68	94052.62	616.68	380294.22
612.56	375229.75	720.15	518616.02
460.03	211627.60	782.62	612494.06
375.67	141127.95	728.84	531207.75
424.35	180072.92	979.52	959459.43
360.48	129945.83	977.33	955173.93
Sum = 3325.82	1258085.10	Sum = 7463.55	5382184.64

So, E(Xi,1²) = Sum(Xi,1²)/n = 1258085.10/11 = 114371.37

Variance(Xi,1) = E(Xi,1²) - (E(Xi,1))² = (114371.37) - (302.35)² = 22955.85

Similarly, E(Xi,2²) = Sum(Xi,2²)/n = 5382184.64/11 = 489289.51

Variance(Xi,2) = E(Xi,2²) - (E(Xi,2))² = (489289.51) - (678.50)² = 28927.26

Finally, correlation coefficient, r = (17509.41)/(22955.85*28927.26)^{1/2 =} 17509.41/25769.16 = 0.68 (approx).