Question

Toyota company prides themselves on customer service. they have been trying to determine exactly how long it takes, from start to finish, to buy a car at their dealerships. they have determined that the two parts of the transaction (showroom and service) follow the normal model. showroom has a mean time of 3.5 hours with a standard deviation of 1.5 hours. service has an average time of 2 hours with a standard deviation of 0.5 hours.

a) What is the mean and standard deviation of the difference between the showroom and service average waiting time.

b) What is the probability that it will take a customer longer during the service portion of the transaction.

c) Why does the standard deviation always increase when we add or subtract the means of two distributions.

Answer 1

(a)

(i)

For showroom :

1 = 3.5

1 = 1.5

For Service:

2 = 2

2 = 0.5

the mean of the difference between the showroom and service average waiting time: = 3.5 - 2 = 1.5

(ii)

the standard deviation of the difference between the showroom and service average waiting time =

(b)

To find P(X<0):

Z = -1.5/1.5811

= - 0.9487

By Technology, Cumulative Area Under Standard Normal Curve = 0.1714

So,

the probability that it will take a customer longer during the service portion of the transaction = 0.1714

(c)

The standard deviation always increases when we add or subtract the means of two distributions because adding or subtracting two variables increases the overall variability in the outcomes as seen from the following formula independent random variables X and Y::

Var(XY) = Var(X) + Var(Y)

When the variables X and Y are independent, the vectors X and Y are orthogonal. The standard deviation of the sum or difference of the variables X and Y is the hypotenuse of the right triangle. By Pythagorean Theorem, the hypotenuse is always greater than the sides X and Y.

Thus, we prove that the standard deviation always increases when we add or subtract the means of two distributions.