Question

Carlson Mining, Inc. has 12 large pumps that pump water from the mines it operates.
Management has been concerned lately about the amount of money required to
repair malfunctioning pumps. These costs are sums over and above the amounts
spent for routine maintenance. When asked what factors they thought might affect
the mean monthly repair costs, employees suggested the age of the pumps and the
number of hours of operation. The following information was collected.
Choose the relevant Summary Output to answer the questions. Note that the
mean monthly repair costs are in dollars, so when questions ask about repair
costs, make sure that your answers reflect that.
Mean Weekly Hours of Operation over Past Year	Age of Pump at First of Year (months)	Mean Monthly Repair Costs over Past Year
28	80	$1,049
26	48	$1,095
15	27	$882
12	2	$447
16	13	$715
21	55	$542
13	30	$415
21	35	$454
16	36	$495
12	13	$370
18	28	$448
18	49	$509

Develop a linear regression equation to predict the mean monthly repair costs using
the age of the pumps.
How much of the variability in the mean monthly repair costs is explained by the
age of the pumps?
Estimate the mean monthly repair costs for a 32 month old pump. Be careful! Round
your answer like the original data. ROUND CORRECTLY AND INCLUDE
THE UNITS!

Answer 1

Sum of X = 416
Sum of Y = 7421
Mean X = 34.6667
Mean Y = 618.4167
Sum of squares (SS_X) = 4984.6667
Sum of products (SP) = 31486.6667

Regression Equation = ŷ = bX + a

b = SP/SS_X = 31486.67/4984.67 = 6.3167

a = M_Y - bM_X = 618.42 - (6.32*34.67) = 399.4376

ŷ = 6.3167X + 399.4376

X Values
∑ = 416
Mean = 34.667
∑(X - M_x)² = SS_x = 4984.667

Y Values
∑ = 7421
Mean = 618.417
∑(Y - M_y)² = SS_y = 712928.917

X and Y Combined
N = 12
∑(X - M_x)(Y - M_y) = 31486.667

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 31486.667 / √((4984.667)(712928.917)) = 0.5282

So r^2=0.5282^2=0.2790

So 27.90% variability in the mean monthly repair costs is explained by the age of the pumps.

For x=32,

ŷ = (6.3167*32) + 399.4376=601.572=602