Part One: Pearson Correlation

Suppose we follow a 65-year-old woman with low bone density for 8 years; she is treated with alendronate throughout the life of the study. Bone density is measured at various times including at baseline (time = 0 months). We are interested in the strength of the association between time from baseline (the independent variable or X, measured in months) and bone density (the dependent variable or Y, measured in gm/cm²). The data are displayed in the table below:

Visit (i)	Time from bassline (months) (xi)	Bone density Lumbar spine (gm/cm2) (yi)	X-Xbar	(X-Xbar)2	Y-Ybar	(Y-Ybar)2	(X-Xbar)(Y-Ybar)
1	0	0.797	-46.875000	2197.26563	-0.039250	0.001541	1.839844
2	8	0.806	-38.875000	1511.26563	-0.030250	0.000915	1.175969
3	18	0.817	-28.875000	833.76563	-0.019250	0.000371	0.555844
4	48	0.825	1.125000	1.26563	-0.011250	0.000127	-0.012656
5	64	0.837	17.125000	293.26563	0.000750	0.000001	0.012844
6	66	0.841	19.125000	365.76563	0.004750	0.000023	0.090844
7	79	0.886	32.125000	1032.01563	0.049750	0.002475	1.598219
8	92	0.881	45.125000	2036.26563	0.044750	0.002003	2.019344
Sum				8270.87500		0.007454	7.280250