Question

Alumni donations are an important source of revenue for universities. One fundraiser is trying create a model to predict alumni giving rate as a function of 1) graduation rate, 2) percenas smaller class sizes and 3) student faculty ratio. The hypothesized model is y-Bo+B,x, + B₂x₂ + Bx + 6 In this equation, y is alumni giving percentage lie 201 is graduation percentage (1.80) percentage of classes with less than 20 students (ie. 50), and xs is student/faculty ratio A regression analysis of 48 national universities resulted in the following output: age (i.e. 202 less than 20 on analysis of Regression Statistics Multiple R 0.836624531 R Square 0.699940606 Adjusted R Square 0.679482011 Standard Error 7.609724781 Observations 48 ANOVA dr Significance F 1.43233E-11 Regression Residual Total 3 44 47 SS MS F 5943.531072 1981.177 34.21255 2547.948094 57.90791 8491.479167 Coefficients Standard Error Star P-value Lower 95% Upper 95% Intercept -20.7201343 17.52136501 -1.18256 0.243333 -56.03212475 14.5918561 0.748182799 0.165959959 4.508213 4.8E-05 0.413712483 1.08265312 % of Classes Under 20 0.029040648 0.139321322 0.208444 0.835844 -0.251743023 0.30982432 Student Faculty Ratio - 1.19201069 0.386723104 -3.08234 0.003538 -1.971399888 -0.4126215
1) The quality measure we use to check the validity of the model is
a. Significance F b. Coefficient of determination >
c. SSE d. Standard error >
e. none of the above.
2) Which of the following is the correct interpretation of the graduation rate" coefficient?
a. For each point increase in alumni giving percentage, graduation rate increases by .75%
b. For each point increase in graduation percentage, alumni giving rate increases by 41%
c. For each point increase in alumni giving percentage, graduation rate increases by 41%
d. For each point increase in graduation percentage, alumni giving rate increases by .75%
e. none of the above.
3) How do you interpret the value of the intercept term bo?
a. It is the loss a university must absorb if alumni funds dry up
b. It is the historical minimum alumni giving percentage
c. It is the average amount of funds these 48 universities raise yearly
d. It cannot be interpreted because zero graduation rate, zero small class size and zero student faculty ratio are outside the ranges used to develop the regression model.
e. none of the above.
4) As student/faculty ratio increases, how does this affect giving rate and why
a. Cannot tell without a specific student/faculty percentage
b. Alumni giving rate improves since the sign of the coefficient is negative
c. Alumni giving rate drops since the sign of the coefficient is negative
d. Must perform a correlation analysis to determine the relation between the two variables
e. none of the above.
5) Based on a t-test and at 5% level of significance, we can conclude that the coefficient of the following independent variable(s) is
0: a. Graduation rate
b. % of classes under 20
c. Student/faculty ratio
d. Graduation rate & student/faculty ratio
e. none of the above.
6) The coefficient of determination for this regression implies
a. that the independent variables are positively correlated
b. the value is too high and therefore the model is not valid
c. approximately 70% of the change in alumni giving rate is explained by the change in the independent variables
d. we can reject that the coefficients are all zero since the value is greater than a
e. none of the above.
7) UTILIZING THE CONCLUSION IN QUESTION ) ABOVE, we predict that a university with an 85% graduation rate, 52% small classes and a student/faculty ratio of 12 will have an alumni giving rate of
a. 30.08
b. 28.57
c. 44.39
d. 35.16
e. none of the above.

Answer 1

Answer:

Step-by-step explanation:

Hello!

Given the variables

Y: Alumni donations

X₁: Graduation rate

X₂: percentage of classes with less than 20 students

X₃: student/faculty ratio

The multiple regression model is:

Y= β₀ + β₁X₁ + β₂X₂ + β₃X₃ + ε

1)

The correct option is

e. none of the above.

The measurement used to determine the validity of the test is the coefficient of determination R².

2)

The general interpretation of a regression coefficient βi is:

"quantifies the modification of the mean of Y when the response variable Xi increases one unit while all the other response variables remain constant"

^β₁= b₁= 0.748% represents the "graduation rate coefficient"

It is the modification of the average alumni donations when the graduation rate increases one unit while the  percentage of classes with less than 20 students and the student/faculty ratio remain constant.

Correct option: d. For each point increase in graduation percentage, alumni giving rate increases by .75%

3)

The intercept (b₀) is the value of the average of Y when all response variables are zero.

b₀= -20.72%

Using a general interpretation: -20.72% is the average alumni donation when the graduation rate, percentage of classes with less than 20 students and student/faculty ratio are zero.

Of course this interpretation is theoretical, meaning, it does not have any practical application.

The correct option is:

e. none of the above.

4)

To determine how does and increase of the student/faculty ratio you have to take a look at the estimation of the regression coefficient that corresponds to that variable:

b₃= -1.1920

This ratio means that every time the student/faculty ratio increases one unit, the giving rate decreases 1.192%

Correct option:

c. Alumni giving rate drops since the sign of the coefficient is negative

5)

I'll use the given p-values to check the significance of each individual hypothesis test. The decision rule for all of them is as follows:

If p-value ≤ α, reject the null hypothesis.

If p-value > α, don't reject the null hypothesis.

I)

H₀: β₁ = 0

H₁: β₁ ≠ 0

α: 0.05

p-value: 4.8E-5 ⇒ Reject the null hypothesis.

II)

H₀: β₂ = 0

H₁: β₂ ≠ 0

α: 0.05

p-value: 0.835844 ⇒ Do not reject the null hypothesis.

III)

H₀: β₃ = 0

H₁: β₃ ≠ 0

α: 0.05

p-value: 0.003538 ⇒ Reject the null hypothesis.

Correct option:

d. Graduation rate & student/faculty ratio

6)

R²= 0.6999⇒ 69.99%

Correct interpretation:

c. approximately 70% of the change in alumni giving rate is explained by the change in the independent variables

7)

^Y= -20.72 + 0.748X₁ + 0.029X₂ - 1.192X₃

For X₁= 85, X₂= 52 and X₃= 12

^Y= -20.72 + 0.748*85 + 0.029*52 - 1.192*12= 30.064

Correct option

a. 30.08

I hope it helps!