Consider the table below. Construct a confidence for the mean predicted when X=7 at the 0.10 level significance. *

 

X 5, 3, 6, 3, 4, 4, 6, 8

Y 13, 15,7,12,13,11, 9, 5

The 90 % confidence interval for predicted value of Y when X=7 is ( 3.1911, 10.6533).

Step-by-step explanation

First of all we will estimate the linear regression line for the given data.

The equation of linear regression line is,

?y=β0?+β1?∗x+??

??β1?=n∗∑x2−(∑x)2n∗∑xy−∑x∗∑y????

???β1?=8∗211−(39)28∗378−39∗85???

?β1?=−1.7425?

?β0?=y??−β1?∗x??

?β0?=10.625+1.7425∗4.875?

?β0?=19.1198?

Thus the estimated regression line is,

?y^?=19.1198−1.7425∗x?

The estimated value of y when x = 7 is,

?y^?=19.1198−1.7425∗7?

?y^?=6.9222?

The confidence interval for the predicted values is given by,

?y^?±tcritical?∗Se??

?tcritical?=? ?t0.10,6?? = 1.94318

And,

?Se?=Syx?∗1+1/n+SSx?(x0?−x?)2???

???Se?=1.6579∗1+1/8+20.875(7−4.875)2?????

?Se?=1.9201?

6.9222 ?±? 1.94318 * 1.9201

?

Thus 90 % confidence interval for predicted value is ( 3.1911, 10.6533).

please see the attached file for the complete solution.