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Chapter 5: Problem 24

Data on high school GPA \((x)\) and first-year college GPA ( \(y\) ) collected from a southeastern public research university can be summarized as follows ("First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students," Journal of College Student Development \([1999]: 599-605):\) $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ a. Find the equation of the least-squares regression line. b. Interpret the value of \(b\), the slope of the least-squares line, in the context of this problem. c. What first-year GPA would you predict for a student with a \(4.0\) high school GPA?

Short Answer

Expert verified
a. The least-squares regression line is \(y = -0.106 + 0.578x\). b. The slope \(b = 0.578\) means for each one-point increase in high school GPA \(x\), we predict the first-year college GPA \(y\) to increase by 0.578. c. The predicted GPA for a student with a 4.0 high school GPA is 2.206.

Step by step solution

01

Calculate the Parameters

First, calculate the slope \(b\) and intercept \(a\) of the least-square regression line using the provided summary statistics. The formulas used for the parameters are:\(b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\)\(a = \frac{\sum y - b(\sum x)}{n}\)Substituting the given values we get:\(b = \frac{2600*27918 - 9620*7436}{2600*36168 - 9620^2} = 0.578\)\(a = \frac{7436 - 0.578*9620}{2600} = -0.106\)
02

Form the Regression Line

The least squares regression line is given by the equation \(y = a + bx\)Substitute the values calculated in step 1 into this equation to get the regression line. So, the least squares regression line is \(y = -0.106 + 0.578x\)
03

Interpret the Slope

The slope \(b = 0.578\) can be interpreted as follows: for each one-unit increase in the high school GPA \(x\), we predict the first-year college GPA \(y\) to increase by 0.578. Therefore, for a student with 1 higher point in high school GPA, we expect them to score 0.578 higher in their first-year college GPA, all other things being equal.
04

Predict the GPA

To predict the first-year GPA for a student with a 4.0 high school GPA, substitute \(x=4.0\) into the regression line equation.So, the predicted GPA is\(y = -0.106 + 0.578 * 4.0 = 2.206\)

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Key Concepts

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

High School GPA
The high school Grade Point Average (GPA) is a critical measure widely used in educational systems to gauge a student's overall academic performance. Typically measured on a four-point scale, with 4.0 being the highest score representing an A average, a high school GPA is calculated by averaging the final grades of all completed courses.

Understanding the influence of high school GPA on first-year college success is important for several reasons. It offers colleges an indicator to identify potentially successful students, and it helps students to set realistic expectations for their academic future. Scholars and educational authorities often conduct statistical analyses to investigate the high school GPA's predictive power in relation to college performance.
First-Year College GPA
Much like the high school GPA, the first-year college GPA is a significant indicator of a student鈥檚 academic achievement during their freshman year of college. It can be impacted by multiple factors, such as the difficulty of college courses, adaptation to college life, and personal discipline.

The transition from high school to college often entails new educational challenges, so tracking the first-year college GPA allows educators and academic advisors to provide appropriate support for students who may struggle during this crucial period. Studies investigating the relationship between high school GPA and college GPA help in understanding which factors contribute to college success.
Predictive Modeling
Predictive modeling involves using statistics to predict outcomes. In the context of educational predictive modeling, we use previous data, such as high school GPA, to forecast future academic success, such as first-year college GPA. These types of models can guide academic advising, admissions decisions, and tailored student support services.

A least-squares regression line is a common predictive model in this realm. This statistical tool helps to predict one variable based on the presence of another. For instance, by analyzing the relationship between high school GPA and first-year college GPA, educators can use the least-squares regression line to anticipate a student鈥檚 college performance.
Statistical Analysis
Statistical analysis involves collecting, reviewing, and presenting data in a way that extracts useful information. In educational research, it helps to understand the relationships between different academic indicators.

The process of calculating a least-squares regression line, as in the provided exercise, includes computing the slope and the intercept through formulas derived from summary statistics. These calculations help us understand the strength and type of correlation between high school GPA and first-year college GPA. The slope, for instance, tells us how much we expect the college GPA to increase for each one-point increase in the high school GPA, holding all other variables constant. Such insights are invaluable for students and educators looking to enhance educational outcomes.

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Most popular questions from this chapter

In the article "Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana" (Journal of Herpetology [1999]: \(100-105\) ), 14 female salamanders were studied. Using regression, the researchers predicted \(y=\) clutch size (number of salamander eggs) from \(x=\) snout-vent length (in centimeters) as follows: $$ \hat{y}=-147+6.175 x $$ For the salamanders in the study, the range of snout-vent lengths was approximately 30 to \(70 \mathrm{~cm}\). a. What is the value of the \(y\) intercept of the least-squares line? What is the value of the slope of the least-squares line? Interpret the slope in the context of this problem. b. Would you be reluctant to predict the clutch size when snout-vent length is \(22 \mathrm{~cm}\) ? Explain.

An auction house released a list of 25 recently sold paintings. Eight artists were represented in these sales. The sale price of each painting appears on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist \((x)\) and sale price \((y)\) ? Why or why not?

The article "Air Pollution and Medical Care Use by Older Americans" (Health Affairs [2002]: 207-214) gave data on a measure of pollution (in micrograms of particulate matter per cubic meter of air) and the cost of medical care per person over age 65 for six geographical regions of the United States: $$ \begin{array}{lcc} \text { Region } & \text { Pollution } & \text { Cost of Medical Care } \\ \hline \text { North } & 30.0 & 915 \\ \text { Upper South } & 31.8 & 891 \\ \text { Deep South } & 32.1 & 968 \\ \text { West South } & 26.8 & 972 \\ \text { Big Sky } & 30.4 & 952 \\ \text { West } & 40.0 & 899 \\ & & \\ \hline \end{array} $$ a. Construct a scatterplot of the data. Describe any interesting features of the scatterplot. b. Find the equation of the least-squares line describing the relationship between \(y=\) medical cost and \(x=\) pollution. c. Is the slope of the least-squares line positive or negative? Is this consistent with your description of the relationship in Part (a)? d. Do the scatterplot and the equation of the least-squares line support the researchers' conclusion that elderly people who live in more polluted areas have higher medical costs? Explain.

The article "Characterization of Highway Runoff in Austin, Texas, Area" (Journal of Environmental Engineering \([1998]: 131-137\) ) gave a scatterplot, along with the least-squares line for \(x=\) rainfall volume (in cubic meters) and \(y=\) runoff volume (in cubic meters), for a particular location. The following data were read from the plot in the paper: $$ \begin{array}{rrrrrrrrr} x & 5 & 12 & 14 & 17 & 23 & 30 & 40 & 47 \\ y & 4 & 10 & 13 & 15 & 15 & 25 & 27 & 46 \\ x & 55 & 67 & 72 & 81 & 96 & 112 & 127 & \\ y & 38 & 46 & 53 & 70 & 82 & 99 & 100 & \end{array} $$ a. Does a scatterplot of the data suggest a linear relationship between \(x\) and \(y\) ? b. Calculate the slope and intercept of the least-squares line. c. Compute an estimate of the average runoff volume when rainfall volume is 80 . d. Compute the residuals, and construct a residual plot. Are there any features of the plot that indicate that a line is not an appropriate description of the relationship between \(x\) and \(y\) ? Explain.

Peak heart rate (beats per minute) was determined both during a shuttle run and during a 300 -yard run for a sample of \(n=10\) individuals with Down syndrome ("Heart Rate Responses to Two Field Exercise Tests by Adolescents and Young Adults with Down Syndrome," Adapted Physical Activity Quarterly [1995]: 43-51), resulting in the following data: $$ \begin{array}{llllllll} \text { Shuttle } & 168 & 168 & 188 & 172 & 184 & 176 & 192 \\ \text { 300-yd } & 184 & 192 & 200 & 192 & 188 & 180 & 182 \\ \text { Shuttle } & 172 & 188 & 180 & & & & \\ \text { 300-yd } & 188 & 196 & 196 & & & & \end{array} $$ a. Construct a scatterplot of the data. What does the scatterplot suggest about the nature of the relationship between the two variables? b. With \(x=\) shuttle run peak rate and \(y=300\) -yd run peak rate, calculate \(r\). Is the value of \(r\) consistent with your answer in Part (a)? c. With \(x=300\) -yd peak rate and \(y=\) shuttle run peak rate, how does the value of \(r\) compare to what you calculated in Part (b)?
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