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Chapter 15: Problem 46

The SAT and ACT exams are often used to predict a student's first-term college grade point average (GPA). Different formulas are used for different colleges and majors. Suppose that a student is applying to State U with an intended major in civil engineering. Also suppose that for this college and this major, the following model is used to predict first term GPA. $$ \begin{aligned} G P A &=a+b(A C T) \\ a &=0.5 \\ b &=0.1 \end{aligned} $$ a. In this context, what would be the appropriate interpretation of the value of \(a\) ? b. In this context, what would be the appropriate interpretation of the value of \(b ?\)

Short Answer

Expert verified
a. The value of \(a = 0.5\) represents the baseline predicted GPA for a student with an ACT score of 0, as a Civil Engineering major at State U. b. The value of \(b = 0.1\) represents the expected increase in the first-term GPA of a student for each one-point increase in their ACT score, in the context of a Civil Engineering major at State U.

Step by step solution

01

a. Interpretation of the value of \(a\).

In this model, \(a = 0.5\) represents the y-intercept or the predicted GPA when the ACT score is 0. It can be interpreted as the baseline GPA that a student with an ACT score of 0 is predicted to have as a Civil Engineering major at State U.
02

b. Interpretation of the value of \(b\).

In this model, \(b = 0.1\) represents the slope of the linear relationship between ACT score and GPA. It can be interpreted as the expected increase in the first-term GPA of the student for each one-point increase in their ACT score. In other words, a student with an ACT score of (x+1) is predicted to have a first-term GPA that is 0.1 points higher than a student with an ACT score of x, in the context of Civil Engineering major at State U.

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Key Concepts

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

SAT and ACT Exams
Standardized tests such as the SAT (Scholastic Assessment Test) and the ACT (American College Testing) are crucial steps in the college admission process in the United States. They aim to measure a student's readiness for college by assessing various skills, including mathematics, critical reading, and writing.

The results from SAT and ACT exams are not just metrics for college admissions but are also used as indicators of academic success. Colleges often use these scores to predict academic performance, including a student's GPA during their first term at college. Institutions rely on historical data to draw correlations between standardized test scores and college GPAs, helping them in decision-making for admissions and placement.
Linear Regression Model
The linear regression model is a statistical method used to predict the value of a dependent variable based on the value of one or more independent variables. This model assumes that there is a linear relationship between the inputs and outputs.

In educational statistics, a linear regression equation can be used to predict a student鈥檚 GPA based on their standardized test scores like the SAT or ACT. The equation typically has the form of Y = a + bX, where Y is the predicted value鈥� in this case, GPA鈥� a is the y-intercept, and b is the slope that represents the rate of change in GPA for every unit change in the test score.
Educational Statistics
Educational statistics involve the application of statistical methods to data from educational settings to inform policy and practice. It encompasses everything from the analysis of standardized test scores to measure educational achievement to the prediction of students' academic success.

Specifically, in predicting college GPA, educational statisticians would use data such as SAT or ACT scores to create predictive models. These models help to understand the strength and nature of the relationship between test scores and GPA, guiding both admissions strategies and student support services.

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Most popular questions from this chapter

15.19 Acrylamide is a chemical that is sometimes found in cooked starchy foods and which is thought to increase the risk of certain kinds of cancer. The paper "A Statistical Regression Model for the Estimation of Acrylamide Concentrations in French Fries for Excess Lifetime Cancer Risk Assessment" (Food and Chemical Toxicology [2012]: \(3867-3876\) ) describes a study to investigate the effect of frying time (in seconds) and acrylamide concentration (in micrograms per kilogram) in french fries. The data in the accompanying table are approximate values read from a graph that appeared in the paper. \begin{tabular}{|cc|} \hline Frying Time & Acrylamide Concentration \\ \hline 150 & 155 \\ 240 & 120 \\ 240 & 190 \\ 270 & 185 \\ 300 & 140 \\ 300 & 270 \\ \hline \end{tabular} a. For these data, the estimated regression line for predicting \(y=\) acrylamide concentration based on \(x=\) frying time is \(y=87+0.359 x\). What is an estimate of the average change in acrylamide concentration associated with a 1-second increase in frying time? b. What would you predict for acrylamide concentration for a frying time of 250 seconds? c. Use the given Minitab output to decide if there is convincing evidence of a useful linear relationship between acrylamide concentration and frying time. You may assume that the necessary conditions have been met. R-sq \(\begin{array}{cc}\text { R-sq(adj) } & \text { R-sq(pred) } \\ 0.00 \% & 0.00 \%\end{array}\) \(\mathrm{S}\) 3 \(\mathrm{q}\) \(8 \%\) Coefficients \(\mathrm{K}-\mathrm{Sq}\) \(14.38 \%\) \(\begin{array}{lccccc}\text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } & \text { VIF } \\ \text { Constant } & 87 & 112 & 0.78 & 0.480 & \\ x & 0.359 & 0.438 & 0.82 & 0.459 & 1.00\end{array}\) Regression Equation \(y=87+0.359 x\)

The paper "Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International journal of Poultry Science [2008]: \(85-88\) ) suggests that the simple linear regression model is reasonable for describing the relationship between \(y=\) eggshell thickness (in micrometers) and \(x=\) egg length (mm) for quail eggs. Suppose that the population regression line is \(y=0.135+0.003 x\) and that \(\sigma=0.005 .\) Then, for a fixed \(x\) value, \(y\) has a normal distribution with mean \(0.135+0.003 x\) and standard deviation 0.005 . a. What is the mean eggshell thickness for quail eggs that are \(15 \mathrm{~mm}\) in length? For quail eggs that are \(17 \mathrm{~mm}\) in length? b. What is the probability that a quail egg with a length of \(15 \mathrm{~mm}\) will have a shell thickness that is greater than \(0.18 \mu \mathrm{m} ?\) c. Approximately what proportion of quail eggs of length \(14 \mathrm{~mm}\) have a shell thickness of greater than \(0.175 ?\) Less than \(0.178 ?\)

Do taller adults make more money? The authors of the paper "Stature and Status: Height, Ability, and Labor Market Outcomes" (Journal of Political Economics [2008]: 499-532) investigated the association between height and earnings. They used the simple linear regression model to describe the relationship between \(x=\) height (in inches) and \(y=\) log(weekly gross earnings in dollars) in a very large sample of men. The logarithm of weekly gross earnings was used because this transformation resulted in a relationship that was approximately linear. The paper reported that the slope of the estimated regression line was \(b=0.023\) and the standard deviation of \(b\) was \(s_{b}=0.004\). Carry out a hypothesis test to decide if there is convincing evidence of a useful linear relationship between height and the logarithm of weekly earnings. You can assume that the basic assumptions of the simple linear regression model are met.

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Consider a test of hypotheses about, \(\beta\) the population slope in a linear regression model. a. If you reject the null hypothesis, \(\beta=0\), what does this mean in terms of a linear relationship between \(x\) and \(y ?\) b. If you fail to reject the null hypothesis, \(\beta=0,\) what does this mean in terms of a linear relationship between \(x\) and \(y ?\)
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