Question

The time required for an automotive center to complete an oil change service on an automobile approximately follows a normal​ distribution, with a mean of 19 minutes and a standard deviation of 2 minutes. (b) If the automotive center does not want to give the discount to more than 7​% of its​ customers, how long should it make the guaranteed time​ limit?

Answer 1

To determine the guaranteed time limit for an oil change service such that the automotive center does not want to give the discount to more than 7% of its customers, you can use the standard normal distribution (z-score).

First, find the z-score associated with the 7th percentile. This is the point on the standard normal distribution below which 7% of the data falls. You can use a standard normal distribution table or a calculator for this. The z-score for the 7th percentile is approximately -1.475.

Now, use the z-score formula to find the corresponding value in the original distribution (the time for the oil change):

Z = (X - μ) / σ

Where:
Z is the z-score (-1.475 in this case)
X is the value we want to find (the time limit)
μ is the mean (19 minutes)
σ is the standard deviation (2 minutes)

Plug in the values:

-1.475 = (X - 19) / 2

Now, solve for X (the time limit):

-1.475 * 2 = X - 19

-2.95 = X - 19

X = -2.95 + 19

X = 16.05

So, the automotive center should make the guaranteed time limit approximately 16.05 minutes to ensure that they do not want to give the discount to more than 7% of its customers. Since it's not practical to have a negative time, you can round this up to 17 minutes.