An article in New York Times discussed the relationship between Scholastic Aptit
ID: 3275980 • Letter: A
Question
An article in New York Times discussed the relationship between Scholastic Aptitude Test (SAT) scores and the test-takers’ family incomes. It commented that the wealthier a student’s family, the higher the SAT score. Another common conjecture is that the student’s high school grade point average (GPA) is a good predictor of the student’s SAT score. In the Data Analysis output below, I used regression to find the least squares fit to model SAT score using GPA and Income. The data set I used had 24 students’ SAT scores, the students’ family Incomes (in $) and high school GPA’s. Please use the output given to answer the following questions. It is included in the answer file.
What is the least squares fit equation to predict SAT using Income and GPA? Please state it in equation form: Y = constant + slope*X1 + slope*X2 where you include the actual constant and slope values.
What is the approximate size of errors when using your equation to predict SAT? Compare this to standard deviation of the SAT scores, which is 84.47. What does that comparison tell you about the usefulness of this model?
What fraction of the variation in SAT is explained using Income and GPA in the least squares equation?
What is the F-statistic of the equation? Please interpret it in the context of this problem. What hypothesis does it test? Be specific. What is its p-value? What is your conclusion with respect to the hypothesis?
Interpret the two slopes (in words, in the context of the problem). Are the slope estimates significant? Explain.
Use the equation to predict the SAT of a student with Income = average of the Incomes ($72,833) and GPA = average of the GPA’s (3.28). What is your prediction?
Explanation / Answer
Using the below output file
What is the least squares fit equation to predict SAT using Income and GPA? Please state it in equation form: Y = constant + slope*X1 + slope*X2 where you include the actual constant and slope values.
The regression equation is formed using the coeffiecients as
Y = 1104.258 +0.0017*Income +150.9*GPA
What is the approximate size of errors when using your equation to predict SAT? Compare this to standard deviation of the SAT scores, which is 84.47. What does that comparison tell you about the usefulness of this model?
The standard error of the model is
What fraction of the variation in SAT is explained using Income and GPA in the least squares equation?
This is explained by the r square value . The r square value is 0.865 , this means that the model is able to explain 86.5% variation in the data
What is the F-statistic of the equation? Please interpret it in the context of this problem. What hypothesis does it test? Be specific. What is its p-value? What is your conclusion with respect to the hypothesis?
The significant F is 7.44E-10
The hypothesis
H0 : The model is not statistically significant
H1 : The model is statistically significant
as the p value is less than 0.05 , hence we caonclude that the model is statisitcally significant
The f stat is the ratio of the mean square error of regression and residual 70963.979/1055.613 = 67.225
Interpret the two slopes (in words, in the context of the problem). Are the slope estimates significant? Explain.
0.0017
This means that for every unit chnage in income the value of Y changes by 0.0017 units
150.992
This means that for every unit change in GPA the value of Y changes by 150.992 units
Use the equation to predict the SAT of a student with Income = average of the Incomes ($72,833) and GPA = average of the GPA’s (3.28). What is your prediction?
Y = 1104.258 +0.0017*Income +150.9*GPA is the regression equation
1104.258 +0.0017*72833 +150.9*3.28 = 1723.02
Regression Statistics Multiple R 0.930 R Square 0.865 Adjusted R Square 0.852 Standard Error 32.490 Observations 24 ANOVA df SS MS F Significance F Regression 2 141927.957 70963.979 67.225 7.44E-10 Residual 21 22167.876 1055.613 Total 23 164095.833 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1104.258 54.752 20.168 3.17E-15 990.394 1218.122 Income 0.0017 0.00025 6.770 1.07E-06 0.001 0.002 GPA 150.992 15.093 10.004 1.92E-09 119.604 182.380Related Questions
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