In a study, the protein absorption (Y) for seven concentration levels (X) of tha
ID: 2928608 • Letter: I
Question
In a study, the protein absorption (Y) for seven concentration levels (X) of that protein were measured:
Conc. Level (Xi)
6
8
10
12
14
16
18
Absorption (Yi)
10
15
18
18
24
22
26
We are interested in fitting the following simple linear regression model:
This can also be written in matrix form as Y = X +
a) Calculate XX, (XX)-1 and XY and then calculate the least squares estimates of 0 and 1.
b) Calculate the variance-covariance matrix of b, and use it to perform a t-test to test the null hypothesis that 1 = 0. Use = 0.05.
Conc. Level (Xi)
6
8
10
12
14
16
18
Absorption (Yi)
10
15
18
18
24
22
26
Explanation / Answer
Y = [10;15;18;18;24;22;26]
Y =
10
15
18
18
24
22
26
>> X =[1 6;1 8;1 10; 1 12; 1 14 ; 1 16; 1 18]
X =
1 6
1 8
1 10
1 12
1 14
1 16
1 18
a)
X'*X
ans =
7 84
84 1120
inv = inv(X'*X)
ans =
1.4286 -0.1071
-0.1071 0.0089
X'*Y
ans =
133
1732
b = inv(X'*X)*X'*Y
ans =
4.4286
1.2143
hence b0 = 4.4286 , b1 = 1.2143
b)
SSE = (Y - X*b)' * (Y - X*b)
SSE =
16.8571
error_Variance = SSE/(length(Y) -2)
error_Variance =
3.3714
variance-covariance matrix of b =
var_covar = error_Variance*inv(X'*X)
var_covar =
4.8163 -0.3612
-0.3612 0.0301
t = 1.2143/sqrt(0.0301)
= 6.9991
since t > critical value
we reject the null
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