Explain why the con?dence interval for the mean value of y/ Microeconomics
1. Learning the mechanics.
(a) Explain why for a particular x-value, the prediction interval for an individual y-value will always be wider than the con?dence interval for a mean value of y.
(b) Explain why the con?dence interval for the mean value of y for a particular x-value, say, xp, gets wider the farther xp is from x¯. What are the implications of this phenomenon for estimation and prediction?
2. Learning the mechanics. A simple linear regression analysis for n = 20 data points produced the following results:
yˆ = 2.1 + 3.4x SSxx = 4.77
x¯ = 2.5 SSyy = 59.21
y¯ = 10.6 SSxy = 16.22
(a) Find SSE and s2.
(b) Find a 95% con?dence interval for E(y) when x = 2.5. Interpret this interval.
(c) Find a 95% con?dence interval for E(y) when x = 2.0. Interpret this interval.
(d) Find a 95% con?dence interval for E(y) when x = 3.0. Interpret this interval.
(e) Examine the widths of the con?dence intervals obtained in parts b, c, and d. What happens to the width of the con?dence interval for E(y) as the value of x moves away from the value of x¯?
(f) Find a 95% prediction interval for a value of y to be observed in the future when x = 3.0. Interpret its value.
