Question

Do students with higher college grade point averages (GPAs) earn more than those graduates with lower GPAs?† Consider the following hypothetical college GPA and salary data (10 years after graduation).

GPA	Salary ($)
2.22	72,000
2.27	48,000
2.57	72,000
2.59	62,000
2.77	88,000
2.85	96,000
3.12	133,000
3.35	130,000
3.66	157,000
3.68	162,000

(a)

Develop a scatter diagram for these data with college GPA as the independent variable.

A scatter diagram has a horizontal axis labeled "GPA" with values from 2 to 4 and a vertical axis labeled "Salary ($)" with values from 0 to 180,000. The scatter diagram has 10 points. A pattern goes down and right from (2.22, 122,000)to (3.68, 8,000). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "GPA" with values from 2 to 4 and a vertical axis labeled "Salary ($)" with values from 0 to 180,000. The scatter diagram has 10 points. A pattern goes down and right from (2.22, 162,000)to (3.68, 48,000). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "GPA" with values from 2 to 4 and a vertical axis labeled "Salary ($)" with values from 0 to 180,000. The scatter diagram has 10 points. A pattern goes up and right from (2.22, 48,000) to (3.68, 162,000). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "GPA" with values from 2 to 4 and a vertical axis labeled "Salary ($)" with values from 0 to 180,000. The scatter diagram has 10 points. A pattern goes up and right from (2.22, 8,000) to (3.68, 122,000). The points are scattered moderately from the pattern.

What does the scatter diagram indicate about the relationship between the two variables?

The scatter diagram indicates no apparent relationship between GPA and salary.The scatter diagram indicates a negative linear relationship between GPA and salary.     The scatter diagram indicates a positive linear relationship between GPA and salary.The scatter diagram indicates a nonlinear relationship between GPA and salary.

(b)

Use these data to develop an estimated regression equation that can be used to predict annual salary 10 years after graduation given college GPA. (Let x = GPA, and let y = salary (in $).Round your numerical values to the nearest integer.)

ŷ =  

(c)

At the 0.05 level of significance, does there appear to be a significant statistical relationship between the two variables? (Use the F test.)

State the null and alternative hypotheses.

H₀: β₀ ≠ 0
H_a: β₀ = 0H₀: β₁ = 0
H_a: β₁ ≠ 0     H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₀ = 0
H_a: β₀ ≠ 0H₀: β₁ ≥ 0
H_a: β₁ < 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Do not reject H₀. There is a significant statistical relationship between GPA and salary.Reject H₀. There is a significant statistical relationship between GPA and salary.      Reject H₀. There is not a significant statistical relationship between GPA and salary. Do not reject H₀. There is not a significant statistical relationship between GPA and salary.

Answer 1

a) Scatter plot:

The scatter diagram indicates a positive linear relationship between GPA and salary.

b)

Ʃx =	29.08
Ʃy =	1020000
Ʃxy =	3153020
Ʃx² =	87.0846
Ʃy² =	118958000000
Sample size, n =	10
x̅ = Ʃx/n = 29.08/10 =	2.908
y̅ = Ʃy/n = 1020000/10 =	102000
SSxx = Ʃx² - (Ʃx)²/n = 87.0846 - (29.08)²/10 =	2.51996
SSyy = Ʃy² - (Ʃy)²/n = 118958000000 - (1020000)²/10 =	14918000000
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 3153020 - (29.08)(1020000)/10 =	186860

Slope, b = SSxy/SSxx = 186860/2.51996 = 74151.971

y-intercept, a = y̅ -b* x̅ = 102000 - (74151.97067)*2.908 = -113633.9

Regression equation :

ŷ = -113634 + (74152) x

-------

c)

Null and alternative hypothesis:

Ho: β₁ = 0

Ha: β₁ ≠ 0

SSE = SSyy -SSxy²/SSxx = 14918000000 - (186860)²/2.51996 = 1061962761.3137

SSR = SSxy²/SSxx = (186860)²/2.51996 = 13856037238.6863

Test statistic:

F = SSR/(SSE/(n-2)) = 13856037238.6863/(1061962761.3137/8) = 104.38

p-value = F.DIST.RT(104.3806, 1, 8) = 0.000

Conclusion:

Reject H₀. There is a significant statistical relationship between GPA and salary.