# Confidence Interval for Linear Regression

I'm in an Econometrics class and we just finished taking our final exams. In the exam there was one particular question regarding confidence intervals that I can't figure out. I'm sure there's a simple answer but it's going to drive me crazy not being able to figure it out. Any insight/direction would be greatly appreciated!

The Question

A linear regression has been estimated for predicting the value of a house given several factors describing the house.

$$\widehat price = \underset{(8,452)}{245,595} + \underset{(0.23)}{1.20(sqrft - 2200)} + \underset{(824)}{2465(bdrms - 4)} + \underset{(76)}{1256(location-1)}$$

Where:

$sqrft$ is the square footage of the house

$bdrms$ is the number of bedrooms in the house

$location$ is a binary variable indicating the home resides in a favorable location

Given the estimated model, calculate the 90% confidence interval for the estimated price of a house with 2200 square feet, 4 bedrooms, and resides in a favorable location.

Assume the relevant $t$ statistic is $1.65$.

Given this question and my basic knowledge of statistics I naively assumed that given the specification of the house each coefficient would be multiplied by zero leaving us with only the intercept.

Assuming that, we would then calculate a confidence interval using the equation:

$$C.I. = \hat\beta \pm t \times se(\hat\beta)$$

In this particular case:

$$C.I. = 245,595\pm(1.65\times8,452)$$

However the resulting confidence interval was not a valid choice. So, my limited knowledge of stats now exhausted, I gave up and made my best SWAG :(

I believe what you need is actually a prediction interval since you are bounding it for one point, not for the mean.

In that case, you should adjust your model to the s.e. prediction, which should be

$$C.I._{proposed} = \hat\beta \pm t \times se(\hat\beta)$$
Is problematic in that $\hat\beta$ is a vector, not a scalar. If we multiply by the input vector $X$ (where $X=[1\ sqrft\ bdrms\ location]$ we get
$$C.I. = X\hat\beta \pm t \times X(se(\hat\beta))$$