Part A

An economist has measured people's annual salary (in thousands of dollars) and their years of relevant job experience, thinking that a linear relationship between them might exist.

Let the proposed regression relationship between Salary and experience be as follows: E(Salary) = beta subscript 0 space plus space beta subscript 1 space cross times Years of Experience

and assume the output from running the regression is as follows:

Call:

lm(formula = Salary ~ Year, data = Income)

Residuals:

Min 1Q Median 3Q Max

-53.650 -20.256 0.127 18.423 65.596

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 31.8387 8.5565 3.721 0.00033***

Years 2.8205 0.3302 8.543 1.74e-13 ***

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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 25.98 on 98 degrees of freedom

Multiple R-squared: 0.4268, Adjusted R-squared: 0.421

F-statistic: 72.98 on 1 and 98 DF, p-value: 1.737e-13

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Residual standard error: 8.044 on 445 degrees of freedom

Multiple R-squared: 0.6914, Adjusted R-squared: 0.6886

F-statistic: 249.2 on 4 and 445 DF, p-value: < 2.2e-16

If we wished to conduct a hypothesis test as to whether there is a linear relationship between salary and years of experience, what are the correct null and alternate hypotheses?

Answers:

a.

H subscript 0 : space beta subscript 0 space equals space 0 H subscript 1 : space beta subscript 0 greater than space 0

b.

H subscript 0 : space beta subscript 0 space equals space 0 H subscript 1 : space beta subscript 0 space end subscript not equal to space 0

c.

H subscript 0 space : thin space beta subscript 1 space equals space 0 H subscript 1 : space beta subscript 1 space end subscript space not equal to space 0

d.

H subscript 0 : space beta subscript 1 space equals space 0 H subscript 1 : space beta subscript 1 space greater than space 0

Part B

Using the output in Q1, what is the correct p-value for the test in Q1?

Answers:

a.

0.00033

b.

0.000000000000174

c.

1.74e-13

d.

0.00000393

Part C

What is the fitted regression model from this output in Q1?

Answers:

a.

E(Salary) = 31.8387 + 2.8205 x Years of Experience

b.

E( Years of Experience ) = 2.8205 + 31.8387 x Salary

c.

E( Years of Experience ) = 31.8387 + 2.8205 x Salary

d.

E(Salary) = 2.8205 + 31.8387 x Years of Experience

Part D

Which of the following is a correct statement regarding r squared ?

Answers:

a.

r squared space equals space 0.4268 meaning that Years of Experience explains 42.68 percent sign of the variability in Salary.

b.

r squared space equals space 0.00033 meaning that Years of Experience explains 0.033 percent sign of the variability in Salary.

c.

r squared space equals space 0.00033 and because 0.00033 space less than space 0.05 we reject H subscript 0 and accept H subscript 1 at the 5% level of significance, ie we conclude there is a significant linear relationship between Salary and Years of Experience.

d.

r squared space equals space 0.4268 and because 0.4268 space greater than space 0.05 we do not reject H subscript 0 at the 5% level of significance, ie we conclude there is no significant linear relationship between Salary and Years of Experience.