Question

A prominent university conducted a survey on the effect of part-time work on student grade point average (GPA). Let x be the hours worked per week and y the GPA for the year. A summary of the results is below. What can the university conclude with an α of 0.05?

n = 21

sigmay = 55

,sigma x = 520

sigmay² = 171

, aigmax² = 15288

sigmayx = 1275

, sigma ( y − ŷ2)  = 24

a) Compute the quantities below.

_Bhat0 =  , Bhat =

What GPA is predicted when a students works 9 hours a week?

b) Compute the appropriate test statistic(s) for H₁: β < 0.

Critical value =  ; Test statistic =

Decision:  

---Select---

Reject H0

Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).

If not appropriate, input and/or select "na" below.

Effect size =  ;  

---Select---

na

trivial effect

small effect

medium effect

large effect

d) Make an interpretation based on the results.

More hours of part-time work significantly predicts a higher GPA.

More hours of part-time work significantly predicts a lower GPA.    

Part-time work does not significantly predict GPA.

Answer 1

Ʃx = 520

Ʃy = 55

Ʃxy = 1275

Ʃx² = 15288

Ʃy² = 171

Sample size, n = 21

x̅ = Ʃx/n = 520/21 = 24.7619048

y̅ = Ʃy/n = 55/21 = 2.61904762

SSxx = Ʃx² - (Ʃx)²/n = 15288 - (520)²/21 = 2411.80952

SSyy = Ʃy² - (Ʃy)²/n = 171 - (55)²/21 = 26.952381

SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 1275 - (520)(55)/21 = -86.9047619

SSE = Ʃ(y-ŷ)² = 24 (given)

--

a) Slope, β = SSxy/SSxx = -86.90476/2411.80952 = -0.036033 = -0.036

y-intercept, β₀ = y̅ - β* x̅ = 2.61905 - (-0.03603)*24.7619 = 3.5112936 = 3.5113

Regression equation :

ŷ = 3.5113 + (-0.036) x

Predicted value of y at x = 9

ŷ = 3.5113 + (-0.036) * 9 = 3.187

b)

Null and alternative hypothesis:

Ho: β₁ = 0

Ha: β₁ < 0

α = 0.05

df = n-2 = 19

Critical value, t_c = ABS(T.INV(0.05, 19)) = -1.729

Standard error, se = √(SSE/(n-2)) = √(24/(21-2)) = 1.12390

Test statistic:

t = b/(se/√SSxx) = -1.58

Conclusion:

Fail to reject the null hypothesis.

--

c) effect size:

r² = (SSxy)²/(SSxx*SSyy) = (-86.90476)²/(2411.80952*26.95238) = 0.1162

small effect

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d) Part-time work does not significantly predict GPA.