Professor I.M. Mean likes to use averages. When fitting a regression model \(y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}\) using the \(N=6\) observations in Table 2.4 from Exercise 2.3, \(\left(y_{i}, x_{i}\right)\), Professor Mean calculates the sample means (averages) of \(\left(y_{i}, x_{i}\right)\) for the first three and second three observations in the data \(\left(\bar{y}_{1}=\sum_{i=1}^{3} y_{i} / 3, \bar{x}_{1}=\sum_{i=1}^{3} x_{i} / 3\right)\) and \(\left(\bar{y}_{2}=\sum_{i=4}^{6} y_{i} / 3, \bar{x}_{2}=\sum_{i=4}^{6} x_{i} / 3\right)\). Then Dr. Mean's estimator of the slope is \(\hat{\beta}_{2, \text { mean }}=\left(\bar{y}_{2}-\bar{y}_{1}\right) /\left(\bar{x}_{2}-\bar{x}_{1}\right)\) and the Dr. Mean intercept estimator is \(\hat{\beta}_{1, \text { mean }}=\bar{y}-\hat{\beta}_{2, \text { mean }} \bar{x}\), where \((\bar{y}, \bar{x})\) are the sample means using all the data. You may use a spreadsheet or other software to carry out tedious calculations.

a. Calculate \(\hat{\beta}_{1, \text { mean }}\) and \(\hat{\beta}_{2, \text { mean }}\). Plot the data, and the fitted line \(\hat{y}_{i, \text { mean }}=\hat{\beta}_{1, \text { mean }}+\hat{\beta}_{2, \text { mean }} x_{i}\).

b. Calculate the residuals \(\hat{e}_{i, \text { mean }}=y_{i}-\hat{y}_{i, \text { mean }}=y_{i}-\left(\hat{\beta}_{1, \text { mean }}+\hat{\beta}_{2, \text { mean }} x_{i}\right)\). Find \(\sum_{i=1}^{6} \hat{e}_{i, \text { mean }}\), and \(\sum_{i=1}^{6} x_{i} \hat{e}_{i, \text { mean }}\).

c. Compare the results in

(b) to the corresponding values based on the least squares regression estimates. See Exercise 2.3.

d. Compute \(\sum_{i=1}^{6} \hat{e}_{i, \text { mean }}^{2}\). Is this value larger or smaller than the sum of squared least squares residuals in Exercise 2.3(d)?

Data From Exercise 2.3:-

Graph the following observations of \(x\) and \(y\) on graph paper.

a. Using a ruler, draw a line that fits through the data. Measure the slope and intercept of the line you have drawn.

b. Use formulas (2.7) and (2.8) to compute, using only a hand calculator, the least squares estimates of the slope and the intercept. Plot this line on your graph.

c. Obtain the sample means \(\bar{y}=\sum y_{i} / N\) and \(\bar{x}=\sum x_{i} / N\). Obtain the predicted value of \(y\) for \(x=\bar{x}\) and plot it on your graph. What do you observe about this predicted value?

d. Using the least squares estimates from (b), compute the least squares residuals \(\hat{e}_{i}\).

e. Find their sum, \(\sum \hat{e}_{i}\), and their sum of squared values, \(\sum \hat{e}_{i}^{2}\).

f. Calculate \(\sum x_{i} \hat{e}_{i}\).

Data From Formula 2.7 and 2.8:-

Transcribed Image Text:

TABLE 2.4 Exercise 2.3 Data x 1 2 3 y 6 4 11 49 69 17 53 13