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Chapter 13: Problem 2

Does study help GPA? For the Georgia Student Survey file on the book's website, the prediction equation relating \(y=\) college \(\mathrm{GPA}\) to \(x_{1}=\) high school GPA and \(x_{2}=\) study time (hours per day), is \(\hat{y}=1.13+\) \(0.643 x_{1}+0.0078 x_{2}\) a. Find the predicted college GPA of a student who has a high school GPA of 3.5 and who studies three hours a day. b. For students with fixed study time, what is the change in predicted college GPA when high school GPA increases from 3.0 to \(4.0 ?\)

Short Answer

Expert verified
a. Predicted college GPA is 3.40. b. College GPA increases by 0.643 points.

Step by step solution

01

Define the Prediction Equation

The prediction equation provided is \( \hat{y} = 1.13 + 0.643x_1 + 0.0078x_2 \), where \( \hat{y} \) is the predicted college GPA, \( x_1 \) is the high school GPA, and \( x_2 \) is the study time in hours per day. We will use this to find specific predicted values.
02

Calculate Predicted College GPA for Part (a)

Substitute \( x_1 = 3.5 \) and \( x_2 = 3 \) into the equation:\[\hat{y} = 1.13 + 0.643 \times 3.5 + 0.0078 \times 3\]Calculate each term:- \( 0.643 \times 3.5 = 2.2505 \)- \( 0.0078 \times 3 = 0.0234 \)Add these to the constant 1.13:\[\hat{y} = 1.13 + 2.2505 + 0.0234 = 3.4039\]Thus, the predicted college GPA is approximately 3.40.
03

Determine the Effect of High School GPA Increase for Part (b)

When \( x_1 \) changes from 3.0 to 4.0, the change in \( x_1 \) is \( \Delta x_1 = 1.0 \). For a fixed \( x_2 \), the change in predicted GPA \( \Delta \hat{y} \) due to \( \Delta x_1 \) is calculated using the coefficient of \( x_1 \):\[\Delta \hat{y} = 0.643 \times \Delta x_1 = 0.643 \times 1.0 = 0.643\]Thus, the predicted college GPA increases by 0.643 points.

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Key Concepts

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

Prediction Equation
In linear regression, a prediction equation is used to estimate the value of a dependent variable based on one or more independent variables. The equation typically takes the form of a straight line. In this case, the prediction equation is \( \hat{y} = 1.13 + 0.643x_1 + 0.0078x_2 \). Here, \( \hat{y} \) represents the predicted college GPA, \( x_1 \) stands for the high school GPA, and \( x_2 \) is the study time, measured in hours spent per day.
  • The constant term \( 1.13 \) indicates the baseline college GPA prediction when both high school GPA and study time are zero, although such a scenario doesn't make practical sense in this context.
  • The coefficient \( 0.643 \) next to \( x_1 \) signifies how much the college GPA is expected to increase for each additional point in high school GPA, assuming study time remains constant.
  • The coefficient \( 0.0078 \) illustrates how much the college GPA is expected to increase for each additional hour of study per day, keeping the high school GPA stable.
The prediction equation thus provides a straightforward method to estimate future outcomes based on known input values.
College GPA
College GPA, or Grade Point Average, is a measure of a student's academic performance during their time at a college or university. It is typically calculated on a scale from 0 to 4.0, with 4.0 representing perfect grades and 0 indicating failing grades.
  • A higher GPA indicates better academic performance and can significantly impact opportunities for scholarships, graduate school acceptance, and employment.
  • The prediction equation helps estimate a student's college GPA based on their high school GPA and study habits. This preemptive measure can guide students in adjusting their study habits to achieve their desired GPA goals.
Using the prediction equation aids students in foreseeing their potential academic performance and making informed decisions to enhance their learning strategies.
High School GPA
High school GPA is a crucial factor in predicting college success, as reflected in our prediction equation. It's a cumulative reflection of a student's academic achievements and is used by colleges for admissions decisions. It also plays a significant part in our prediction equation for college GPA.
  • In this particular exercise, the high school GPA impacts the college GPA prediction by a factor of \( 0.643 \). This implies that each additional point in high school GPA increases the predicted college GPA by 0.643, assuming study time is constant.
  • This relationship underscores the importance of maintaining a strong academic record throughout high school, as it has a lasting effect on future educational success.
These insights reveal the weight of high school performance in shaping academic opportunities and achievements in college.
Study Time
Study time, measured in hours per day, is a vital yet often underestimated factor influencing academic success in college. In our exercise, study time is the second independent variable in the prediction equation.
  • Each additional hour of study per day contributes to an anticipated increase of 0.0078 in the college GPA when high school GPA remains unchanged. Though the coefficient is smaller than that of high school GPA, consistent study habits can lead to gradual improvement over time.
  • Effective study strategies, combined with sufficient study time, can optimize student performance, thus resulting in a higher predicted college GPA.
The inclusion of study time in the prediction equation serves to remind students of the importance of dedicated study periods as part of their academic routine for achieving desired GPA outcomes.

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Most popular questions from this chapter

Voting and income A logistic regression model describes how the probability of voting for the Republican candidate in a presidential election depends on \(x,\) the voter's total family income (in thousands of dollars) in the previous year. The prediction equation for a particular sample is $$ \hat{p}=\frac{e^{-1.00+\operatorname{an} 2 x}}{1+e^{-1.00+0.02 x}} $$ Find the estimated probability of voting for the Republican candidate when (a) income \(=\$ 10,000\), (b) income \(=\$ 100,000 .\) Describe how the probability seems to depend on income.

Controlling has an effect The slope of \(x_{1}\) is not the same for multiple linear regression of \(y\) on \(x_{1}\) and \(x_{2}\) as compared to simple linear regression of \(y\) on \(x_{1},\) where \(x_{1}\) is the only predictor. Explain why you would expect this to be true. Does the statement change when \(x_{1}\) and \(x_{2}\) are uncorrelated?

Suppose you fit a straight-line regression model to \(y=\) number of hours worked (excluding time spent on household chores) and \(x=\) age of the subject. Values of \(y\) in the sample tend to be quite large for young adults and for elderly people, and they tend to be lower for other people. Sketch what you would expect to observe for (a) the scatterplot of \(x\) and \(y\) and (b) a plot of the residuals against the values of age.

Predicting visitor satisfaction For all the restaurants in a city, the prediction equation for \(y=\) average monthly visitor satisfaction rating (range \(0-4.0\) where \(0=\) very poor and \(4=\) very good) and \(x_{1}=\) the monthly food quality score given by the food inspection authority (range \(0-4.0\) where \(0=\) very poor and \(4=\) very good) and \(x_{2}=\) the number of visitors in a month is \(\hat{y}=0.35+0.55 x_{1}+0.0015 x_{2}\) a. Find the predicted average monthly visitor satisfaction rating for a restaurant having (i) a monthly food quality score of 4.0 and 800 visitors in a month and (ii) a monthly food quality score of 2.0 and 200 visitors in a month. b. For restaurants with \(x_{2}=500\), show that \(\hat{y}=1.10+0.55 x_{1}\) c. For restaurants with \(x_{2}=600,\) show that \(\hat{y}=1.25+0.55 x_{1}\). Thus, compared to part b, the slope for \(x_{1}\) is still \(0.55,\) and increasing \(x_{2}\) by 100 (from 500 to 600 ) shifts the intercept upward by \(100 \times\left(\right.\) slope for \(\left.x_{2}\right)=100(0.0015)=0.15\) units.

When we use multiple regression, what is the purpose of performing a residual analysis? Why is it better to work with standardized residuals than unstandardized residuals to detect outliers?
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