Using SAS programming studio mmass age 106 43 106 41 97 47 113 46 96 45 119 41 9

ID: 3355583 • Letter: U

Question

Using SAS programming studio

mmass age 106 43 106 41 97 47 113 46 96 45 119 41 92 47 112 41 92 48 102 48 107 42 107 47 102 43 115 44 101 42 87 55 91 57 97 56 82 59 78 57 95 54 98 53 94 52 96 53 100 54 84 60 70 59 104 51 76 59 93 57 73 68 73 63 76 60 80 63 84 63 71 64 64 66 88 65 79 60 88 65 73 65 74 69 76 61 87 70 70 68 69 78 54 78 62 78 78 72 65 70 64 73 74 76 87 78 63 78 82 71 80 75 52 77 56 76 70 72 74 76 In what units is o exp b. Estimate 2 and . To explore this rela- age group, 7. Muscle mass. A person's muscle mass is expected to decrease with age. To explore tionship in women, a nutritionist randomly selected 15 women from each 10-year beginning with age 40 and ending with age 79. The results follow; X is age, and y is a me of muscle mass. Assume that first-order regression model (1.1) is appropriate. 58 59 60 72 76 56 70 74 2 43 Xi: Y 106 106 97... 41 47 76 a. Obtain the estimated regression function. Plot the estimated regression function and the data Does a linear regression function appear to give a good fit here? Does your plot support the anticipation that muscle mass decreases with age? b. Obtain the following: (1) a point estimate of the difference in the mean muscle mass for women differing in age by one year, (2) a point estimate of the mean muscle mass for women aged X = 60 years, (3) the value of the residual for the eighth case, (4) a point estimate of

Explanation / Answer

Given

xi = age and yi = measure of muscle mass of the ith woman under study, the regression back-up theory is as follows:

Back-up theory

The linear regression model Y = 0 + 1X + , ………………………………………..(1)

where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2.

Estimated Regression of Y on X is given by: Y = 0cap + 1capX, ………………………….(2)

where

1cap = Sxy/Sxx and 0cap = Ybar – 1cap.Xbar..……………………………………………..(3)

Mean X = Xbar = (1/n)sum of xi ………………………………………….……………….(4)

Mean Y = Ybar = (1/n)sum of yi ………………………………………….……………….(5)

Sxx = sum of (xi – Xbar)2 …………………………………………………..………………………………..(6)

Syy = sum of (yi – Ybar)2 …………………………………………………..………………………………..(7)

Sxy = sum of {(xi – Xbar)(yi – Ybar)} …………………………………………………………………….………(8)

All above sums are over i = 1, 2, …., n

n = sample size ………………………………………………………………………………(9)

Estimate of 2 is given by s2 = (Syy – 1cap2Sxx)/(n - 2)…………………………………..(10)

Summary of Excel Calculations:

xbar

ybar

84.967

Sxx

8210.9833

Syy

15501.9333

Sxy

-9771.0333

1cap

-1.1899955

0cap

156.346564

s^2

66.8008189

Answers

a) Regression function: ycap = 156.345 – 1.190x ANSWER

b) (1): Point estimate of difference in mean muscle mass for women differing in age by one year is equivalent to the difference in mean muscle mass for women per unit change (i.e., 1 year) in age. And that is precisely the interpretation of the slope coefficient estimate,

1cap = 1.190 (decrease) ANSWER 1

b) (2): Point estimate of the mean muscle mass for women aged 60 years is ycap at x = 60

= 156.345 – (1.190 x 60) = 84.945 ANSWER 2

b) (3): Value of residual of eighth case = y8 - y8cap

= 112 – {156.345 – (1.190 x 41)} = 4.445 ANSWER 3

b) (4): Point estimate of 2 = s2 = 66.8008 ANSWER 4