Question

The following table contains the \(A C T\) scores and the \(G P A\) (grade point average) for eight college students. Grade point average is based on a four- point scale and has been rounded to one digit after the decimal. $$\begin{array}{|ccc|} \hline \text { Student } & G P A & A C T \\ \hline 1 & 2.8 & 21 \\ 2 & 3.4 & 24 \\ 3 & 3.0 & 26 \\ 4 & 3.5 & 27 \\ 5 & 3.6 & 29 \\ 6 & 3.0 & 25 \\ 7 & 2.7 & 25 \\ 8 & 3.7 & 30 \\ \hline \end{array}$$ i. Estimate the relationship between \(G P A\) and \(A C T\) using \(0 \mathrm{LS}\); that is, obtain the intercept and slope estimates in the equation $$\widehat{G P A}=\widehat{\beta}_{0}+\widehat{\beta}_{1} A C T$$ Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the \(G P A\) predicted to be if the \(A C T\) score is increased by five points? ii. Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum to zero. iii. What is the predicted value of \(G P A\) when \(A C T=20 ?\) iv. How much of the variation in \(G P A\) for these eight students is explained by \(A C T\) ? Explain.

Short answer

The predicted GPA for ACT=20 is determined by \(\widehat{GPA} = \widehat{\beta}_{0} + \widehat{\beta}_{1} \times 20\). The direction of the relationship is given by the sign of \(\widehat{\beta}_{1}\). The intercept suggests GPA at ACT=0 but isn't practical.

Step-by-step solution

Calculate the Mean of GPA and ACT

First, we calculate the means of GPA and ACT scores. To find the mean, add all the values in the column and divide by the number of students (8 in this case). \[ \text{Mean GPA} = \frac{2.8 + 3.4 + 3.0 + 3.5 + 3.6 + 3.0 + 2.7 + 3.7}{8} \approx 3.2125 \] \[ \text{Mean ACT} = \frac{21 + 24 + 26 + 27 + 29 + 25 + 25 + 30}{8} \approx 25.875 \]

Calculate the Slope Estimate

Using the formula for the slope in a simple linear regression, we have: \[ \widehat{\beta}_{1} = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}} \] Substitute the values from each student's (GPA, ACT) scores. Calculate the sums and finally the slope \(\widehat{\beta}_{1}\).

Calculate the Intercept Estimate

The intercept \(\widehat{\beta}_{0}\) can be calculated using the formula: \[ \widehat{\beta}_{0} = \bar{Y} - \widehat{\beta}_{1} \cdot \bar{X} \] Substitute \(\bar{Y}\) as the mean GPA and \(\bar{X}\) as the mean ACT from Step 1, along with \(\widehat{\beta}_{1}\) from Step 2 to find the intercept.

Determine the Relationship and Interpretation of Intercept

The slope \(\widehat{\beta}_{1}\) determines the direction and strength of the relationship between ACT scores and GPA. A positive \(\widehat{\beta}_{1}\) indicates a positive relationship. While the intercept \(\widehat{\beta}_{0}\) indicates the estimated GPA when ACT score is 0, it may not have practical meaning if ACT score cannot be 0 in reality.

Calculate GPA Increase for a Five-point Increase in ACT

Multiply the slope \(\widehat{\beta}_{1}\) by the change in ACT (5) to find how much the GPA changes.\[ \Delta \widehat{GPA} = 5 \cdot \widehat{\beta}_{1} \]

Calculate Fitted Values and Residuals

For each student, calculate the fitted GPA using the regression equation: \[ \widehat{GPA}_i = \widehat{\beta}_{0} + \widehat{\beta}_{1} \cdot ACT_i \] Then, compute the residual for each student as:\[ \text{Residual}_i = GPA_i - \widehat{GPA}_i \] Sum the residuals and verify they are (approximately) zero.

Predict GPA for ACT=20

Use the regression equation to estimate the GPA for an ACT score of 20:\[ \widehat{GPA} = \widehat{\beta}_{0} + \widehat{\beta}_{1} \times 20 \]

Calculate Variation Explained by ACT

The proportion of variation in GPA explained by ACT scores is the coefficient of determination \(R^2\). It can be calculated as:\[ R^2 = \frac{SS_{Regression}}{SS_{Total}} \] Where \(SS_{Regression}\) is the sum of squares due to regression and \(SS_{Total}\) is the total sum of squares. However, in a simple linear regression, \(R^2\) can also be inferred directly from the correlation between GPA and ACT.

Key concepts

GPA

Grade Point Average, or GPA, is a crucial aspect of academic assessment in schools and colleges. It is a numerical representation of a student's academic performance. Typically measured on a 4.0 scale, GPA provides an averaged summary of a student's grades over a specific period, such as a semester or an academic year.

• A 4.0 GPA indicates perfect academic performance, equating to an "A" in every class.


• Lower GPAs such as 2.7 or 3.0 reflect various combinations of grades ranging from "B" to "C".

Understanding the GPA is essential when analyzing student performance because it directly impacts future educational and career opportunities. GPA often serves as an entry criterion for higher education programs and scholarships. Here, it is used to establish a relationship with ACT scores through a simple linear regression analysis.

ACT Scores

ACT scores are standardized test results used widely in the United States to assess high school students' readiness for college. They cover four main academic areas: English, mathematics, reading, and science reasoning. The composite score ranges from 1 to 36.

• Higher ACT scores typically correlate with higher academic aptitude.


• Many colleges consider ACT scores alongside GPA in their admission processes.

In this exercise, ACT scores serve as an independent variable in the regression analysis. Researchers often explore correlations between ACT scores and GPAs to predict future academic success in college settings. Observations like these help in understanding the potential relationship between standardized testing and academic performance.

Slope and Intercept

In Simple Linear Regression, two key components define the relationship between variables: the slope and the intercept. The slope (\(\widehat{\beta}_{1}\)) represents the change in the dependent variable (GPA) for a one-unit change in the independent variable (ACT score).

• A positive slope indicates that as ACT scores increase, GPA is also likely to increase.


• A negative slope would imply a decrease in GPA with increasing ACT scores, although this is less common.

The intercept (\(\widehat{\beta}_{0}\)) denotes the expected value of the dependent variable when the independent variable is zero. In practical terms, this might not always have meaningful interpretation, especially since zero ACT scores aren't feasible. Nevertheless, intercept can serve as a baseline measure in the regression equation. Calculations involving slope and intercept allow us to predict trends, such as estimating a GPA increase when ACT scores increase by a specific amount.

Residuals

Residuals in regression analysis represent the differences between observed values and those predicted by the regression equation. A residual for each observation can be calculated by subtracting the predicted GPA (\(\widehat{GPA}_i\)) from the actual GPA.

• Residuals illustrate the amount of error present in the predictions.


• A positive residual implies that the actual GPA is higher than predicted, while a negative one suggests it is lower.

Summing residuals should approximate zero in an accurately fitted model. Analyzing these residuals helps identify how well the model fits the data. Large residuals may indicate outliers or that a simple linear model is not well-suited to the data.

Coefficient of Determination

The Coefficient of Determination, commonly represented as \(R^2\),is a measure that explains the proportion of variance in the dependent variable (GPA) that can be predicted from the independent variable (ACT score).

• An \(R^2\) value of 1 indicates perfect prediction, meaning 100% of the variance in GPA is explained by ACT scores.


• An \(R^2\) value of 0 indicates no predictive capacity.


• In practice, \(R^2\) values between 0 and 1 provide insight into the strength of the relationship.

In this exercise, calculating \(R^2\) gives us an idea of how well ACT scores predict GPA among the eight students. High \(R^2\) values suggest strong predictive power, which can be useful when interpreting the model's reliability and relevance.