Using the regression model for prediction a particular value of y

The second major use of a regression model is to predict a particular value of y for a given value of x, say . For example, we may want to predict the food expenditure of a randomly selected household with a monthly income of $3000. In this case, we are not interested in the mean food expenditure of all households with a monthly income of $3000 but in the food expenditure of one particular household with a monthly income of $3000. This predicted value of y is denoted by . To predict a single value of y for from estimated sample regression line, we use the value of as a point estimate of . Using the estimated regression line, we find for x =30 as

Thus, based on our regression line, the point estimate for the food expenditure of a given household with a monthly income of $3000 is $906.74 per month.

Different regression lines estimated by using different samples of seven households each taken from the same population will give different values of the point estimator for the predicted value of y for x =30.Hence, a confidence interval constructed for based on one sample will give a more reliable estimate of than will a point estimate. The confidence interval constructed for is more commonly called a prediction interval.

Suppose that the population regression model is

and that the standard regression assumptions hold, and that the are normally distributed. Let a and b be the least squares estimates of and .

It can be shown that the following are intervals:

1. For the forecast of the single value resulting for at a given , the prediction interval is

2. For the forecast of the conditional expectation, , the confidence interval is

where

and .

Example:

For the data on incomes and food expenditures of seven households, find

a) 99% prediction interval for the predicted food expenditure for a single household with a monthly income of $3500;

b) Obtain a 99% confidence interval for the expected food expenditure for all households with a monthly income of $3000.

Solution:

a) The point estimate of the predicted food expenditure for is given by

Using data from the previous chapters

; ; and

Hence, the 99% prediction interval for for is

to

Thus, with 99% confidence we can state that the predicted food expenditure of a household with a monthly income of $3500 is between $606.00 and $1471.68.

b) Once again, the point estimate of the expected food expenditure for is

Hence, the 99% confidence interval for is

to

Thus, with 99% confidence we can state that the mean food expenditure for all households with monthly income of $3500 is between $873.61 and $1204.07.

As we can observe, the interval in part a) is much wider than the one for the mean value of y for calculated in part b) . This is always true. The prediction interval for predicting a single value of y is always larger than the confidence interval for estimating the mean value of y for a certain value of x.

Дата добавления: 2015-08-05