Question

Suppose a local university researcher wants to build a linear model that predicts the freshman year GPA of incoming students based on high school SAT scores. The researcher randomly selects a sample of 40 sophomore students at the university and gathers their freshman year GPA data and the high school SAT score reported on each of their college applications. He produces a scatterplot with SAT scores on the horizontal axis and GPA on the vertical axis. The data has a linear correlation coefficient of 0.506701. Additional sample statistics are summarized in the table below. Sample Variable Sample standard deviation mean Variable description high school SAT score freshman year GPA x= 1504.291401 sx 105.782904 у y = 3.240805 Sy = 0.441205 r=0.506701 slope = 0.002113 Determine the y-intercept, a, of the least-squares regression line for this data. Give your answer precise to at least four decimal places. a =​

Answer 1

a = 0.0617

Explanation:

Given That:

Relationship between GPA and SAT Scores :

Linear correlation Coefficient (r) = 0.506701

x= 1504.291401

Sx = 105.782904

y = 3.240805

Sy = 0.441205

The intercept (a) :

a = y - b(x)

b = slope of the regression line

b = r(Sy/Sx) ;

Hence,

b = 0.506701(0.441205 / 105.782904)

b = 0.506701(0.0041708)

b = 0.0021133485

Hence,

a = y - b(x)

a = 3.240805 - 0.0021133485(1504.291401)

a = 3.240805 - 3.1790919758662485

a = 0.0617130241337515

a = 0.0617

Hence, intercept = 0.0617