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Chapter 13: Problem 11

In exercise \(1,\) the following estimated regression equation based on 10 observations was presented. \\[\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}\\] The values of SST and SSR are 6724.125 and \(6216.375,\) respectively. a. Find SSE. b. Compute \(R^{2}\) c. Compute \(R_{\mathrm{a}}^{2}\) d. Comment on the goodncss of fit.

Short Answer

Expert verified
SSE is 507.75; \(R^2\) is 0.924; Adjusted \(R^2\) is 0.9023. It's a good fit.

Step by step solution

01

Identify the Given Values

We have the following values: \(SST = 6724.125\), \(SSR = 6216.375\). We need to find \(SSE\), \(R^2\), and \(R_{a}^{2}\).
02

Calculate SSE

The formula for SSE (Sum of Squares due to Error) is given by: \[SSE = SST - SSR\]Plugging in the given values, we get:\[SSE = 6724.125 - 6216.375 = 507.75\]
03

Calculate R Squared (R^2)

The formula for \(R^2\) is defined as:\[R^2 = \frac{SSR}{SST}\]Substitute \(SSR\) and \(SST\) into the formula:\[R^2 = \frac{6216.375}{6724.125} \approx 0.924\]
04

Calculate Adjusted R Squared (Ra^2)

The formula for the adjusted \(R^2\) is given by:\[R_{a}^{2} = 1 - \left(\frac{(1-R^2) \times (n-1)}{n-k-1}\right)\]Where \(n\) is the number of observations (10), and \(k\) is the number of independent variables (2). First, find \(1-R^2\):\[1-R^2 = 1 - 0.924 = 0.076\]Next, substitute the values into the adjusted \(R^2\) formula:\[R_{a}^{2} = 1 - \left(\frac{0.076 \times (10-1)}{10-2-1}\right) = 1 - \left(\frac{0.684}{7}\right) = 1 - 0.0977 = 0.9023\]
05

Comment on the Goodness of Fit

The \(R^2\) value of approximately 0.924 indicates a very good fit, meaning about 92.4% of the variability in the response variable is explained by the model. The adjusted \(R^2\) value of 0.902 also indicates strong explanatory power while considering the number of predictors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

SST
Sum of Squares Total (SST) is a crucial part of regression analysis. It helps us understand the total variability in the data. SST is calculated by summing up the squares of the differences between each actual observation and the overall mean of the observations. This total variability is divided into two parts: the variability explained by the regression model and the variability not explained, which is error.
  • SST Formula: The formula for SST is: \[ SST = \sum (y_i - \bar{y})^2 \] where \( y_i \) is each individual observation and \( \bar{y} \) is the mean of all observations.
  • Importance: SST helps determine how much of the total variation in the dependent variable can be elucidated by the regression equation.
By understanding SST, you lay the groundwork for comprehending other crucial components like SSR and SSE.
SSR
Sum of Squares for Regression (SSR) is another key component in regression analysis. It represents the portion of the total variability (SST) that is explained by the regression model itself. The larger the SSR, the more variability the model explains, indicating a better fit.
  • SSR Formula: SSR is calculated using the formula: \[ SSR = \sum (\hat{y}_i - \bar{y})^2 \] where \( \hat{y}_i \) are the predicted values of the observations using the regression line, and \( \bar{y} \) is the mean of the actual observations.
  • Role: SSR reflects how well the explanatory variables in the model explain the variability in the dependent variable.
The ratio of SSR to SST, expressed as a percentage ( \[(SSR/SST) \times 100\] ), is what gives us the coefficient of determination, or \( R^2 \).
Goodness of Fit
Goodness of Fit is a measure of how well the regression model captures the actual data. It is evaluated using statistical metrics like \( R^2 \), which indicates the proportion of variance in the dependent variable predictable from the independent variables.
  • \( R^2 \) Value: It鈥檚 calculated as \( R^2 = SSR/SST \), and shows the percentage of the total variability explained by the model. A higher \( R^2 \) value suggests a better fit.
  • Interpretation: An \( R^2 \) value of 0.924 in this context means that 92.4% of the variance in the dependent variable is explained by the model, suggesting a very strong fit.
Alongside \( R^2 \), analyzing residuals and other diagnostic measures can provide a deeper understanding of the model's explanatory power.
Adjusted R Squared
While \( R^2 \) quantifies how well the model explains the data, Adjusted \( R^2 \) takes it a step further. It adjusts \( R^2 \) by considering the number of predictors in the model, providing a more accurate measure when multiple variables are involved.
  • Formula: Adjusted \( R^2 \) is calculated as: \[ R_a^2 = 1 - \left(\frac{(1 - R^2) \times (n - 1)}{n - k - 1}\right) \] where \( n \) is the number of observations, and \( k \) is the number of predictors.
  • Significance: Unlike \( R^2 \), Adjusted \( R^2 \) can increase or decrease when predictors are added or removed. A high Adjusted \( R^2 \) indicates that the model is robust in capturing the variability without overfitting.
In this example, an Adjusted \( R^2 \) of 0.9023 signifies that the model maintains its strong explanatory power even after penalizing for the number of regressors.

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Most popular questions from this chapter

The estimated regression equation for a model involving two independent variables and 10 observations follows. \\[\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}\\] a. Interpret \(b_{1}\) and \(b_{2}\) in this estimated regression equation. b. Estimate \(y\) when \(x_{1}=180\) and \(x_{2}=310\)

A shoe store developed the following estimated regression equation relating sales to inventory investment and advertising expenditures. \\[\hat{y}=25+10 x_{1}+8 x_{2}\\] where \\[\begin{aligned}x_{1} &=\text { inventory investment }(\$ 1000 \mathrm{s}) \\\x_{2} &=\text { advertising expenditures }(\$ 1000 \mathrm{s}) \\\y &=\operatorname{sales}(\$ 1000 \mathrm{s})\end{aligned}\\] a. Estimate sales resulting from a \(\$ 15,000\) investment in inventory and an advertising budget of \(\$ 10,000\) b. Interpret \(b_{1}\) and \(b_{2}\) in this estimated regression equation.

The admissions officer for Clearwater College developed the following estimated regression equation relating the final college GPA (grade point average) to the student's SAT mathematics score and high-school GPA. \\[\hat{y}=-1.41+.0235 x_{1}+.00486 x_{2}\\] where \\[\begin{aligned}x_{1} &=\text { high-school GPA } \\\x_{2} &=\text { SAT mathematics score } \\\y &=\text { final college GPA }\end{aligned}\\] a. Interpret the coefficients in this estimated regression equation. b. Estimate the final college GPA for a student who has a high-school average of 84 and a score of 540 on the SAT mathematics test.

Management proposed the following regression model to predict sales at a fast- food outlet. \\[ y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\epsilon \\] where \\[ \begin{aligned} x_{1} &=\text { number of competitors within 1 mile } \\ x_{2} &=\text { population within 1 mile(1000s) } \\ x_{3} &=\left\\{\begin{array}{l} 1 \text { if drive-up window present } \\ 0 \text { otherwise } \end{array}\right.\\\ y &=\operatorname{sales}(\$ 1000 \mathrm{s}) \end{aligned} \\] The following estimated regression equation was developed after 20 outlets were surveyed. \\[ \hat{y}=10.1-4.2 x_{1}+6.8 x_{2}+15.3 x_{3} \\] a. What is the expected amount of sales attributable to the drive-up window? b. Predict sales for a store with two competitors, a population of 8000 within 1 mile, and no drive-up window. c. Predict sales for a store with one competitor, a population of 3000 within 1 mile, and a drive-up window.

The following estimated regression equation was developed for a model involving two independent variables. \\[\hat{y}=40.7+8.63 x_{1}+2.71 x_{2}\\] After \(x_{2}\) was dropped from the model, the least squares method was used to obtain an estimated regression equation involving only \(x_{1}\) as an independent variable. \\[\hat{y}=42.0+9.01 x_{1}\\] a. Give an interpretation of the coefficient of \(x_{1}\) in both models. b. Could multicollinearity explain why the coefficient of \(x_{1}\) differs in the two models? If so, how?
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