Flavor Deterioration: The results shown below were obtained in a small-scale ex-

Question

Flavor Deterioration: The results shown below were obtained in a small-scale ex- periment to study the relation between o F of storage temperature ( X ) and number of weeks before falovr deterioration of a food product begins to occur ( Y ). Assume that simple linear regression model is applicable. Using matrix methods,

i : 1 2 3 4 5

X : 8 4 0 -4 -8

Y : 7.8 9.0 10.2 11.0 11.7

a) Find ( i ) Y'Y ( ii ) X'Y ( iii ) X'X

b) Estimate the model parameters and write down the fitted regression function.

c) Compute the estimated covariance matrix of the estimated parameters.

d) Obtain the hat matrix H .

e) Obtain the point estimate of _(Y_h) = E ( Y_h ) and the estimated variance of ˆ Y_h , when X_h = 6

Explanation / Answer

Excel formula:

X Y 8 7.8 4 9 0 10.2 -4 11 -8 11.7 a) i) Y'Y 503.77 ii) X'Y -39.2 iii) X'X 160 b) Regression coefficients 1=X'Y / X'X -0.245 0=Y_mean- 0*X_mean 9.94 Regression equation/function: Y= 9.94 - 0.245*X c) Compute the estimated covariance matrix of the estimated parameters Dispersion matrix of X & Y X Y X 40 -9.8 Y -9.8 2.438 d) Hat matrix H= X(X'X)^-1*X' 0.4 0.2 0 -0.2 -0.4 0.2 0.1 0 -0.1 -0.2 0 0 0 0 0 -0.2 -0.1 0 0.1 0.2 -0.4 -0.2 0 0.2 0.4 e) Obtain the point estimate of _(Y_h) = E ( Y_h ) and the estimated variance of ˆ Y_h , when X_h = 6 Y= 9.94 - 0.245*-6 11.41

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