Question

1. Total cholesterol levels (X) and body weights (Y) for six subjects are given below.

cholesterol (X)	0.78	1.04	1.12	1.00	0.75	1.69
body weight (Y)	88.6	74.9	63.7	65.8	93.4	75.4

1. 1. compute i x_iy_i, i x², i y², x¯ and y¯;
2. 2. compute Pearson’s correlation coefficient using the above results;
   3. compute the t-statistic for testing H₀ : corr = 0 vs. H_A : corr ≠ 0.
   4. will you reject H₀ at the 5% level of significance?
   5. find the p-value for the test in c).
   6. produce lists {rank(x₁), . . . , rank(x₆)} and {rank(y₁), . . . , rank(y₆)};
   7. find Spearman’s r_s according to its definition;
   8. find Spearman’s r_s by using rank differences;
   9. using r_s to conduct a similar test as given in part c);
   10. report the p-value for the above test;
   11. find the exact p-value by consulting the attached table. Compare that p-value with the one computed by SAS proc freq.

Answer 1

Find X⋅Y , X² and Y² as it was done in the table below.

X	Y	X⋅Y	X⋅X	Y⋅Y
0.78	88.6	69.108	0.6084	7849.96
1.04	74.9	77.896	1.0816	5610.01
1.12	63.7	71.344	1.2544	4057.69
1.00	65.8	65.8	1	4329.64
0.75	93.4	70.05	0.5625	8723.56
1.69	75.4	127.426	2.8561	5685.16

Find the sum of every column to get:

∑X=6.38 , ∑Y=461.8 , ∑X⋅Y=481.624 , ∑X²=7.363 , ∑Y²=36256.02

x¯ = 6.38 / 6 = 1.0633

y¯ = 461.8 / 6 = 76.9667

( b )

r=−0.4639

( c )