Question

1. Data for 11 states, showing factors affecting SAT scores is summarized in Table (1) State Expenditure S-T Ratio Average Salar Percent Eligible8 Average SAT 6 9 10 4.405 8.9634.778 4.459 4.992 5.443 8.817 7.03 5.718 5.193 17.2 17.6 19.317.1 31.44447.95 32.175 28.934 41.078 34.571 50.045 39.076 32.58832.291 24 18.4 1.16.6 19. 16.3 27 45 29 81 68 48 65 1029 934 944 1005 902 980 908 897 889 854 Table 1: Data affecting SAT scores. Rows 1-4, respectively, indicate the 1997 average expenditure per student (measured in thousands of dollars) the average student-teacher ratio, average teacher salary (measured in thousands of dollars), percent of students eligible to take the SAT. Row 5 provides the average SAT soore for that state (a) Construct a multiple linear regression model of the form y SAT score, y, based upon average expenditure per student, average student-teacher ratio, average teacher salary, and percent of students eligible to take the SAT. In your main writeup indicate the first three rows of of your design matrix as well as your final regression function. (6 points) (zi,22,T3両) that predicts a state's average (b) Calculate the corresponding coefficient of determination, R2. In your writeup indicate the value of R2, the sum of squared errors, SSerror and total sum of squared differences from the mean, SStt (4 points) (c) Data for the State of Michigan is not icluded i Tabl1. Suppose the 1997 average expenditure per student was $6,994, the average student-teacher ratio was 20.1, average teacher salary was $41,895 and, 11% of students were eligible to take the SAT Estimate Michigan's average SAT score based upon the regression model you just constructed. (2 points)

Answer 1

a)

Correlation: Avg. Salary, Avg. Tot. Score

Pearson correlation of Avg. Salary and Avg. Tot. Score = -0.439

P-Value = 0.001

Corr coef = -.0439

There is a weak negative linear relationship between Avg. Salary and Avg. Tot. Score

b)

Scatterplot of Avg. Tot. Score vs Avg. Salary 1150 1100 a 1050 1000 950 900 850 25 30 35 40 45 50 Avg. Salary

Descriptive Statistics: Avg. Salary, Avg. Tot. Score

Variable          N N*    Mean SE Mean StDev Minimum      Q1 Median      Q3 Maximum

Avg. Salary      50   0 34.829    0.840 5.941   25.994 30.890 33.288 38.685   50.045

Avg. Tot. Score 50   0   971.6     10.6   74.9    849.0   902.0   951.5 1038.8   1113.0

Υ.ΣΧ2-Σχ.ΣΧΥ 48579 . 62382.316408-1741.446 . 1682384.304 1164.283 1741.446 446.485795.533 50.62382.316408 n.ΣΧΥ-ΣΧΣΥ n.ΣΧ2 _ (ΣΧ)2 50 . 1682384.304-1741.446 . 48579 50 62382.316408- (1741.446)

У 1164.283-5.533-т

c)

Regression Analysis: Avg. Tot. Score versus Avg. Salary

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 1 52948 52948 11.46 0.001
Avg. Salary 1 52948 52948 11.46 0.001
Error 48 221712 4619
Total 49 274660

Model Summary

S R-sq R-sq(adj) R-sq(pred)
67.9632 19.28% 17.60% 13.94%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 1164.3 57.7 20.17 0.000
Avg. Salary -5.53 1.63 -3.39 0.001 1.00

Regression Equation

Avg. Tot. Score = 1164.3 - 5.53 Avg. Salary

Fits and Diagnostics for Unusual Observations

Avg. Tot.
Obs Score Fit Resid Std Resid
7 914.0 887.4 26.6 0.43 X
40 849.0 996.8 -147.8 -2.21 R

R Large residual
X Unusual X

Residual Plots for Avg. Tot. Score Normal Probability Plot Versus Fits 100 90 50- 10- 14X) 100 14N) 14N) 200 930 1020 Residual Fitted Value Histogram Versus Order 100 100 120 -60 60 120 1 5 10 15 20 25 30 35 40 45 50 Residual Observation Order