The cost of attending your college has once again gone up. Although you have been told that education is investment in human capital, which carries a return of roughly 10% a year, you (and your parents) are not pleased. One of the administrators at your university/college does not make the situation better by telling you that you pay more because the reputation of your institution is better than that of others. To investigate this hypothesis, you collect data randomly for 100 national universities and liberal arts colleges from the 2000-2001 S. News and World Report annual rankings. Next you perform the following regression.
cost 7,311.173,985.20 . Reputation - 0.20 Size +8,406.79. Dpriv
(0.13 2,058.63 (664.58) - 416.38 Dibart - 2,376.51 Dreli ion (2,154.35) (1,121.92) 1,007.56) R' = 0.72 SER 3, 773.35
where Cost is Tuition, Fees, Room and Board in dollars, Reputation is the index used in U.S. News and World Report (based on a survey of university presidents and chief academic officers), which ranges from 1 ("marginal) to 5 ("distinguished"), Size is the number of undergraduate students, and Dpriv, Dlibart, and Dreligion are binary variables indicating whether the institution is private, a liberal arts college, and has a religious affiliation. The numbers in parentheses are heteroskedasticity-robust standard errors
(a) Interpret the results and determine whether or not the coefficients are significantly dif- ferent fromzero. Do the coefficients have the expected sign?
(b) What is the forecasted cost for a liberal arts college, which has no religious affiliation, a size of 1,500 students and a reputation level of 4.5? (All liberal arts colleges are private.)
(c) Suppose that you switch from a private university to a public university, which has a ranking tha's 0.5 lower and 10,000 more students. What is the effect on your cost?
(d) What is the p-value for the null hypothesis that the coefficient on Size is equal to zero? Based on this, should you eliminate the variable from the regression? Why or why not?
(e) You want to test simultaneously the hypotheses ha and bert - 0. Your regression package returns an F-statistic of 1.23. Can you reject the null hypothesis?
(f) Eliminating the Size and Dlibartvariables from your regression, the estimated regression becomes 5,450.35+3,538.84 .
Reputation 1,772.35 (590.49) cost 10,935-70.Dpriv-2, 783.31-Dreligion 575.51) (1,150.57) R20.72 SER- 3,792.68 Why do you think that the effect of attending a private institution has increased now?

Answer:

a) Interpret the results and determine whether or not the coefficients are significantly dif- ferent fromzero. Do the coefficients have the expected sign?

An increase in reputation by one category, increases the cost by

roughly $3,985. The larger the size of the college/university, the

lower the cost. An increase of 10,000 students results in a $2,000

lower cost. Private schools charge roughly $8,406 more than public

schools. A school with a religious affiliation is approximately $2,376

cheaper, presumably due to subsidies, and a liberal arts college also

charges roughly $416 less. There are no observations close to the

origin, so there is no direct interpretation of the intercept. Other than

perhaps the coefficient on liberal arts colleges, all coefficients have

the expected sign.

(b) What is the forecasted cost for a liberal arts college, which has no religious affiliation, a size of 1,500 students and a reputation level of 4.5? (All liberal arts colleges are private.)

7,311.17 + 3,985.20×4.5 – 0.20×1500

+ 8,406.79×1 – 416.38×1 – 2,376.51×0=$ 32,935

(c) Suppose that you switch from a private university to a public university, which has a ranking tha's 0.5 lower and 10,000 more students. What is the effect on your cost?

Roughly $ 12,400. Since over the four years of education, this

implies approximately $50,000, it is a substantial amount of money

for the average household.

(d) What is the p-value for the null hypothesis that the coefficient on Size is equal to zero? Based on this, should you eliminate the variable from the regression? Why or why not?

Using a one-sided alternative hypothesis, the p-value is 6.2 percent. Variables should not be eliminated simply on grounds of a statistical test. The sign of the coefficient is as expected, and its magnitude makes it important. It is best to leave the variable in the regression and let the reader decide whether or not this is convincing evidence that the size of the university matters

(e) You want to test simultaneously the hypotheses ha and bert - 0. Your regression package returns an F-statistic of 1.23. Can you reject the null hypothesis?

The critical value for F2,∞ is 3.00 (5% level) and 4.61 (1% level). Hence you cannot reject the null hypothesis in this case

(f) Eliminating the Size and Dlibartvariables from your regression, the estimated regression becomes 5,450.35+3,538.84 .

Reputation 1,772.35 (590.49) cost 10,935-70.Dpriv-2, 783.31-Dreligion 575.51) (1,150.57) R20.72 SER- 3,792.68 Why do you think that the effect of attending a private institution has increased now?

Step-by-step explanation:

a.An increase in reputation by one category, increases the cost by

roughly $3,985. The larger the size of the college/university, the

lower the cost. An increase of 10,000 students results in a $2,000

lower cost. Private schools charge roughly $8,406 more than public

schools. A school with a religious affiliation is approximately $2,376

cheaper, presumably due to subsidies, and a liberal arts college also

charges roughly $416 less. There are no observations close to the

origin, so there is no direct interpretation of the intercept. Other than

perhaps the coefficient on liberal arts colleges, all coefficients have

the expected sign.

b.7,311.17 + 3,985.20×4.5 – 0.20×1500

+ 8,406.79×1 – 416.38×1 – 2,376.51×0=$ 32,935

c.Roughly $ 12,400. Since over the four years of education, this

implies approximately $50,000, it is a substantial amount of money

for the average household.

d.Using a one-sided alternative hypothesis, the p-value is 6.2 percent. Variables should not be eliminated simply on grounds of a statistical test. The sign of the coefficient is as expected, and its magnitude makes it important. It is best to leave the variable in the regression and let the reader decide whether or not this is convincing evidence that the size of the university matters

e.The critical value for F2,∞ is 3.00 (5% level) and 4.61 (1% level). Hence you cannot reject the null hypothesis in this case

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