Question

1. Consider the linear regression model iid 220 with є, 면 N(0, σ2), i = 1, . . . , n. Let Yh = β0+ßX, be the MLE of the mean at covariate value Xh . (f) Suppose we estimate ơ2 by 82-SSE/(n-2). Derive the distribution for You can use the fact that SSE/σ2 ~ X2-2 without proof. (g) What is a (1-a)100% confidence interval for y? (h) Suppose we observe a new observation Ynet at covariate value X = Xnew. What is a (1-0)100% prediction interval for y ew? (i) Give an intuitive explanation for why the prediction interval from (g) is different than the confi- dence interval from (f)

Answer 1

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AS FOR GIVEN DATA.

Consider the linear regression model with ε¡ 면 N (0, σ2), î 1, . . . , n. Let Yh = β0 + 3X, be the MLE of the mean at covariate value Xh

EXPLANATION ::-

(F) Suppose we estimate ơ2 by s-SSE/(n-2). Derive the distribution for You can use the fact that SSE/σ2 ~ X2-2 without proof

SOL :-

this follow t-distribution with n-2 degree of freedom

as

t = Z/sqrt(V/n)

where Z follow standard normal distribution

V follow chi-square distribution with n degree of freedom

(G) What is a (1-a) 100% confidence interval for y,?

SOL ::-

confidence interval for Yh^

here Xp = Xh

Confidence interval for the conditional mean of the response variable corresponding to xp: rx with df-n-2.

H) Suppose we observe a new observation Ynew at covariate value X-Xnew. What is a (1-α) 100% prediction interval for Ynew

SOL ::-

here x* = x new

prediction interval are

Prediction intervals for specific predicted values A prediction interval for y for a given x is 1 (x* - x)2 )11+-+ x ESI scanned with У t n-2 Sy CamScanner

(I) Give an intuitive explanation for why the prediction interval from (g) is different than the confi- dence interval from (f)

SOL ::-

The difference between a prediction interval and a confidence interval is the standard error.

The standard error for a confidence interval on the mean takes into account the uncertainty due to sampling. The line you computed from your sample will be different from the line that would have been computed if you had the entire population, the standard error takes this uncertainty into account.

The standard error for a prediction interval on an individual observation takes into account the uncertainty due to sampling like above, but also takes into account the variability of the individuals around the predicted mean. The standard error for the prediction interval will be wider than for the confidence interval and hence the prediction interval will be wider than the confidence interval.

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