Question

Given are five observations for two variables,

xi	yi
1	3
2	8
3	5
4	11
5	14

Round your answers to two decimal places.

a. Estimate the standard deviation of ŷ* when x = 4. 2

b. Develop a 95% confidence interval for the expected value of y when x = 4.

c. Estimate the standard deviation of an individual value of y when x = 4.

d. Develop a 95% prediction interval for y when x = 4. If your answer is negative, enter minus (-) sign.

Answer 1

The model that we want to estimate is

where is the intercept and is the slop of the regression line

is a random disturbance

We calculate the following

n=5 is the number of observations

The sample means are

The sum of squares are

The estimate of slope is

The estimate of intercept is

The estimated model is

The sum of square error is

The mean square error is

the standard error of regression is

a. The estimated standard deviation of ŷ* when x = 4 is

b. The predicted value of y when x=4 is

95% confidence level indicates a level of significance . The upper critical value of t is obtained for

The degrees of freedom for t distribution is n-2=5-2=3. Using t tables for df=3 and right tail area =0.025 (total area under 2 tails=0.05) we get

The 95% confidence interval for the expected value of y when x = 4 is

ans: The 95% confidence interval for the expected value of y when x = 4 is [6.64,14.76]

c)  the estimated standard deviation of an individual value of y when x = 4 is

ans: the estimated standard deviation of an individual value of y when x = 4 is 2.66

d)  95% prediction interval for y when x = 4 is

ans: A 95% prediction interval for y when x = 4 is [2.24,19.16]