Using R studio to complete the following problem, please provide the code as wel

Question

Using R studio to complete the following problem, please provide the code as well.

Applied linear regression:

1) George Will is a well-known conservative political commentator. In September 1993, he wrote a column in the Washington Post arguing against the expenditure in public education.

You can read the column here:

https://www.washingtonpost.com/archive/opinions/1993/09/12/meaningless-money-factor/9301c367-cbb6-4086-8705-f785a832eab9/?utm_term=.641d985793e0

In that piece, Mr. Will claimed: ... And the 10 states with the lowest per pupil spending included four - North Dakota, South Dakota, Tennessee, Utah - among the 10 states with the top SAT scores. Only one of the 10 states with the highest per pupil expenditures - Wisconsin - was among the 10 states with the highest SAT scores. New Jersey has the highest per pupil expenditures, an astonishing $10,561, which teachers' unions elsewhere try to use as a negotiating benchmark. New Jersey's rank regarding SAT scores? Thirty-ninth ... The fact that the quality of schools correlates more positively with the quality of the families from which children come to school than it does with education appropriations will have no effect on the teachers unions' insistence that money is the crucial variable. The public education lobby's crumbling last line of defense is the miseducation of the public.

You are going to investigate this claim using data assembled by Guber. This is SAT data assembled for an article on the link between SAT scores and measures of educational expenditures. It is available in R if you load the package faraway and use data("sat"). It will also be available to download here:

https://drive.google.com/file/d/1n-Us9ksNNQS4Af2M2D70QsgM0PJgf_vh/view?usp=sharing

The response variable is SAT score and the explanatory variable used by George Will above is school spending. The data are state level data.

(a) First consider the simple linear regression referred to by George Will: average total SAT score regressed on expenditure per pupil. Also consider other simple linear models for teacher salary, percentage of elgible students taking the SAT, and the student teacher ratio. Describe your findings, and in your writeup, interpret your coeffcients and assess the fit of the regression via residual standard deviation, r^2, and graphically.

(b) Now consider the two variable model of SAT regressed on expenditure and the fraction of students eligible to take the SATs (takers). What do you see? Comment on the regression. Do you see the same effect when you regress SAT on teacher salary and the fraction of eligible students? What about if you include the student-teacher ratio and fraction of eligible students? What are the partial correlations between SAT score and these variables, controlling for the fraction of eligible students taking the exam? (How would you compute this? The correlation between the residuals from the regression of SAT scores on just the fraction of eligible students, and the residuals from the fit of expenditure on this fraction gives the partial correlation of SAT scores and expenditure, controlling for the fraction of eligible students taking the exam.

(c) Now fit a three variable model of SAT on teacher salary, student-teacher ratio, and eligible fraction. Compare this model with the previous ones. Do the coffecients change from the two variable to three variable model? Consider the precision of these estimates when determining if these changes are real.

For this problems, you should use the "pairs" function to examine the scatterplots, and also use coplots to look at relationships of say, expenditure per pupil and total, given an eligible fraction. Also look at the correlations and comment on them.

Explanation / Answer

George Will is a well-known conservative political commentator. In September 1993, he wrote a column in the Washington Post arguing against the expenditure in public education.

You can read the column here:

https://www.washingtonpost.com/archive/opinions/1993/09/12/meaningless-money-factor/9301c367-cbb6-4086-8705-f785a832eab9/?utm_term=.641d985793e0

In that piece, Mr. Will claimed: ... And the 10 states with the lowest per pupil spending included four - North Dakota, South Dakota, Tennessee, Utah - among the 10 states with the top SAT scores. Only one of the 10 states with the highest per pupil expenditures - Wisconsin - was among the 10 states with the highest SAT scores. New Jersey has the highest per pupil expenditures, an astonishing $10,561, which teachers' unions elsewhere try to use as a negotiating benchmark. New Jersey's rank regarding SAT scores? Thirty-ninth ... The fact that the quality of schools correlates more positively with the quality of the families from which children come to school than it does with education appropriations will have no effect on the teachers unions' insistence that money is the crucial variable. The public education lobby's crumbling last line of defense is the miseducation of the public.

You are going to investigate this claim using data assembled by Guber. This is SAT data assembled for an article on the link between SAT scores and measures of educational expenditures.

Here dependent variable is SAT and independent variaables are :

expend

ratio

salary

frac

verbal

math

a) First consider the simple linear regression referred to by George Will: average total SAT score regressed on expenditure per pupil. Also consider other simple linear models for teacher salary, percentage of elgible students taking the SAT, and the student teacher ratio. Describe your findings, and in your writeup, interpret your coeffcients and assess the fit of the regression via residual standard deviation, r^2, and graphically.

Regression of SAT score on expenditure.

We get significant result about expenditure.

Regression of SAT on salary :

Regression of SAT on  percentage of elgible students taking the SAT.

We get significant result about freq.

Regression of SAT on ratio.

Due to this regression we get insignificant result.

b) Now consider the two variable model of SAT regressed on expenditure and the fraction of students eligible to take the SATs (takers). What do you see? Comment on the regression

We get significant result about two variable multiple regression.

Do you see the same effect when you regress SAT on teacher salary and the fraction of eligible students?

We get same result about salary and frac because we get significant result about F-test and t-test.

What are the partial correlations between SAT score and these variables, controlling for the fraction of eligible students taking the exam?

The partial correlation will be 0.9052.

(c) Now fit a three variable model of SAT on teacher salary, student-teacher ratio, and eligible fraction.

Here also we get significant result abot F-test and t-test . All the coefficients are slightly different.

SUMMARY OUTPUT Regression Statistics Multiple R 0.380537 R Square 0.144808 Adjusted R Square 0.126992 Standard Error 69.90851 Observations 50 ANOVA df SS MS F Significance F Regression 1 39722.06 39722.06 8.127774 0.006408 Residual 48 234585.6 4887.2 Total 49 274307.7 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 1089.294 44.38995 24.5392 8.17E-29 1000.042 1178.546 1000.042 1178.546 expend -20.8922 7.328209 -2.85093 0.006408 -35.6265 -6.15782 -35.6265 -6.15782

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