This Final Exam is worth 250 points in total!
In a small-scale regression study, five observations on \(Y\) were obtained corresponding to \(X = 1, 4, 10, 11, 14\). Assume that \(\sigma = 0.6\), \(\beta_0=5\), and \(\beta_1 = 3\).
Draw a residual plot for each of the following cases
An economist studying the relation between household electricity consumption (Y) and the number of rooms in the home (X) employed a simple linear regression model. The following residuals were obtained
\(X_i\) | \(e_i\) |
---|---|
2 | 3.2 |
3 | 2.9 |
4 | -1.7 |
5 | -2.0 |
6 | -2.3 |
7 | -1.2 |
8 | -0.9 |
9 | 0.8 |
10 | 0.7 |
11 | 0.5 |
Plot the residuals \(e_i\) against \(X_i\). What problem appears to be present here? Might a transformation alleviate this problem?
A behavioral scientist said, “I am never sure whether the regression line goes through the origin. Hence, I will not use such a model”. Comment.
Shown below are the number of galleys for a manuscript (X) and the total dollar cost of correcting typographical errors (Y) in a random sample of recent orders handled by a firm specializing in technical manuscripts. Since \(Y\) involves variable costs only, an analyst wished to determine whether the regression-through-the-origin model (4.10) is appropriate for studying the relation between the two variables.
\(X_i\) | \(Y_i\) |
---|---|
7 | 128 |
12 | 213 |
10 | 191 |
10 | 178 |
14 | 250 |
25 | 446 |
30 | 540 |
25 | 457 |
18 | 324 |
10 | 177 |
4 | 75 |
6 | 107 |
For each of the following regression models, indicate whether it is a general linear regression model. If it is not, state whether it can be expressed in the form of (6.7) by a suitable transformation
An analyst wanted to fit the regression model \(Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \beta_3 X_{i3} + \varepsilon_i, i = 1, ..., n\), by the method of least squares when it is known that \(\beta_2 = 4\). How can the analyst obtain the desired fit by using a multiple regression computer program?
The following regression model is being considered in a market research study:
\[Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \beta_3 X_{i1}^2 + \varepsilon_i\]
State the reduced model for testing whether or not
In a regression study, three types of banks were involved, namely, commercial, mutual savings, and savings and loans. Consider the following system of indicator variables for types of bank
Type of Bank | \(X_2\) | \(X_3\) |
---|---|---|
Commercial | 1 | 0 |
Mutual savings | 0 | 1 |
Saving and loans | -1 | -1 |