Question

A recent article in the Cincinnati Enquirer reported that the mean labor cost to repair a heat pump is $90 with a standard deviation of $22. Monte’s Plumbing and Heating Service completed repairs on two heat pumps this morning. The labor cost for the first was $75 and it was $100 for the second. Assume the distribution of labor costs follows the normal probability distribution. Compute z values for each.

Answer 1

z1= -0.6818

z2= 0.45

Explanation:

Mean labor cost to repair the heat pump is μ= $90

standard deviation σ= $22

Labor cost for the first heat pump X_1= $75

labor cost for the second heat pump is X_2= $100

z value for the first heat pump

`z_1= (X_1-\mu)/(\sigma)`

⇒
`z_1= (75-90)/(22)`

= -0.6818

z value for the second heat pump

`z_2= (X_2-\mu)/(\sigma)`

⇒
`z_2= (100-90)/(22)`

= 0.45

Answer 2

Z value for the first pump = -0.68

Z value for the second pump = 0.45

Explanation:

Data provided in the question:

Mean labor cost to repair a heat pump = $90

Standard deviation = $22

Labor cost for the first, X₁ = $75

Labor cost for the Second, X₂ = $100

Now,

Z value for the first pump = [X₁ - Mean] ÷ Standard deviation

thus,

Z value for the first pump = [ $75 - $90] ÷ $22

= - 0.68

Z value for the second pump = [X₁ - Mean] ÷ Standard deviation

thus,

Z value for the second pump = [ $100 - $90] ÷ $22

= 0.45