The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

184 Solutions to Selected Exercises

(b) Least squares the simple linear regression line relating dosage X to sleeping

time Y.

E ( Y | X ) = a + bX , where the intercept bXXYY=∑ni= 11 ()()/ii i− ∧∧− ∑n=

()( Xii−=∑∑ −X∧^2 n== 11 ni X YijnX Y∧∧)[/∑∑ −in==1 1nj X Xijn X()]∧^2

and a = Y ^ − bX ^. From Table 7.6 we see Σ Y = 72 so Y ^ = 72/9 = 8 and Σ X = 8 4

so X ^ = 84/9 = 9.33.

Now b = [6048 − 9(72/9)(84/9)]/[7056 − 9(84/9)(84/9)] = 5376/6272 = 0.857143, and

a = 8 − 0.857(9.33) = 0.00286. The regression line is therefore Y = 0,857 X + 0.00286.

(c) A two - sided 95% confi dence interval for b is obtained by recalling that

SSE =− −∑∑ −Σ()YiiY∧^2 n== 11 nj Y Yijn Y()∧^2. SSyx. =−[()]SSE n/2

and SE b()=∑−Syx. ( [ (Xi X∧)]^2. Also, t = ( b − β )/ SE ( b ) has a t - distribution

with n − 2 degrees of freedom. So the degrees of freedom for this case is 7.

SSE = 5184 − 9(64) = 4608 and SSyx. ==4608 7/ 25 66. and SE b()=

(^) 25 66.[7056 9 9 33−=(.)]..^2 25 66 79 1995 0 324/ =.. Therefore, a
two - sided 95% confi dence interval for b is [ b − 0.324 t 7 (0.975), b + 0.324 t 7 (0.975)].
From the t tables in the appendix, we see that t 7 (0.975) = 2.365. So the confi -
dence interval = [0.857 − 0.324(2.365), 0.857 + 0.324(2.365)] = [0.091, 1.62].
(d) Since 0 is not contained in the interval, we would reject the hypothesis that
β = 0.
Chapter 8

2. In a survey study subjects were asked to report their health as excellent, good,
   poor and very poor. They were also asked to answer whether or not they had

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DOSAGE X (mM/kg)

SLEEPING TIME

Y (hours)

(a) Scatter plot

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