1. USING THE REGRESSION MODEL FOR PREDICTION AND ESTIMATION

2. Form of Confidence Intervals We now use the regression results to calculate point estimates and confidence intervals. Recall that the general form of a confidence interval is: (Point Estimate)  t  /2,DF (Appropriate St’d Deviation) The DF = degrees of freedom for the error term = n-2

3. Point Estimate for Sales in a Single Week What does the model predict that sales will be in a week in which we spend a particular amount, namely $1150? (x p = 1150)

4. Appropriate Standard Deviation For A Single Occurrence We are going to construct a confidence interval for this value. single occurrence PREDICTION INTERVAL.The confidence interval for a single occurrence is called the PREDICTION INTERVAL. The appropriate standard deviation is given by:

5. Prediction Interval for A Single Occurrence

6. 95% Interval Estimate for Sales in A Week in Which $1150 is Spent on Advertising The PRREDICTION INTERVAL is:

7. Point Estimate for the Expected Value (The Long Run Average Value) of Sales When $1150 is Spent on Advertising E(y|x P = 1150) = 46,486.49 + 52.5676(1150) = 106,939.23 That is, the point estimate for the expected value of y equals the point estimate for an individual value; i.e. it equals

8. Appropriate Standard Deviation For The Long Run Average Value We are going to construct a confidence interval for the expected value. long run average valueCONFIDENCE INTERVAL.The confidence interval for the long run average value is called the CONFIDENCE INTERVAL. The appropriate standard deviation is the same as before but without the 1 in the square root:

9. The Long Run Average Value Confidence Interval for The Long Run Average Value of y when x = x P

10. 95% Interval Estimate for the Long Run Average Sales in Weeks in Which $1150 is spent on Advertising

11. Excel =DEVSQ(B2:B11) XPXP =F17+F18*F22=AVERAGE(B2:B11) =I4-TINV(.05,F13)*F7*SQRT(1+1/F8+(I3-I1)^2/I2) Add “$” in F26 by highlighting and pressing F4. Drag down to F29. Change “-” to “+” Delete “1+” Change “-” to “+” and delete “1+”

12. Review Point and Interval Estimates for –The estimated value of y when x = x P PREDICTION INTERVAL –The expected value of y when x = x P CONFIDENCE INTERVAL The difference in the intervals is the 1 under the square root sign Solving for these values –By hand –By Excel