Question

On the last page of this file you will find the Excel output for a multiple linear regression model. The model was built in an attempt to better understand why students at area high schools perform differently on the state high school mathematics test. The average test score for a class of students is what we are trying to predict. In our attempt to understand why these test scores differ, we use 3 independent variables: a rating (0-100) for the quality of the math degree obtained by the instructor, the age of the instructor, and the salary (in thousands) of the instructor.

1. Estimate the average math score for a class of students whose instructor is 52 years old, earns $48,000, and got her degree in a math program rated 72.
2. What percentage of the variations in math scores can be explained by this model?
3. Conduct a test to determine if the model, taken as a whole, provided us with any significant explanation of the differences in math scores. That is, should the model be retained for further analysis?
4. Which of the independent variables appear to be significant to the model? Which appear to be insignificant? What leads you to these conclusions?

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.597512233
R Square	0.357020869
Adjusted R Square	0.303439274
Standard Error	7.724526046
Observations	40
ANOVA
	df	SS	MS	F	Significance F
Regression	3	1192.732105	397.5774	6.663125	0.001076925
Residual	36	2148.058895	59.6683
Total	39	3340.791
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	35.67761801	7.278849159	4.901547	2.03E-05	20.9154278
Math Degree	0.247481581	0.069845662	3.543263	0.001115	0.105828014
Age	0.244830604	0.185213036	1.321886	0.194545	-0.130798841
Income	0.133296712	0.152818937	0.872253	0.388851	-0.176634456

Answer 1

We are already given the summary of regression and ANOVA output of the model.

Here, the average test score for a class is the response variable

And quality rating for the maths degree of instructor, age and the salary of instructor.

The estimated regression line is given as,

Avg test score = 35.67761801 + 0.247481581 Math degree rating + 0.244830604 Age + 0.133296712 Income

To estimate the average math score for a class of students whose instructor is 52 years old, earns $48,000, and got her degree in a math program rated 72.

Put Math degree rating = 72, Age = 52, Income = 48000

Avg test score = 35.67761801 + 0.247481581*72 + 0.244830604*52 + 0.133296712*48000 = 72.6257

The estimated average math score for a class of students whose instructor is 52 years old, earns $48,000, and got her degree in a math program rated 72 is 72.6257.

We are given the value of adjusted R-squared = 0.303439274

We know that 100R² % of the total variation is explained by the model.

Hence, 30.344% of the variations in math scores can be explained by the model.

To test: H₀: The regression model is not significant.

H₁: The regression model is significant.

Test statistic:

Under H₀ test statistic follows F-distribution with (3,36) degrees of freedom.

According to the ANOVA output, F = 6.663125

And P-value for the test is 0.001076925.

For 0.01 level of significance, P-value < 0.01

Hence, we reject H₀ at 0.01 level of significance.

Hence, the regression model is significant. That is the model should be retained for further analysis.

Let us test if the i^th regression coefficient is significant or not.

To test: H₀: =0 versus H₁: 0

Test statistic:

Under H₀ the test statistic follows t distribution with (n-k-1)=36 degrees of freedom.

If the p-value< 0.01 then, we reject H₀ at 0.01 level of significance.

Here, the p-value for math degree is less than 0.01, hence we say that math degree is significant in the model.

But the p-values for Age and salary are greater than 0.01, hence they are insignificant in the model.

Quality of the math degree appear to be significant to the model.

Age and income of instructor appears to be insignificant.

I hope you find the solution helpful. If you have any doubt then feel free to ask in the comment section.

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