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 17 minutes and a standard deviation of 2 minutes.

​(a) The automotive center guarantees customers that the service will take no longer than 20 minutes. If it does take​ longer, the customer will receive the service for​ half-price. What percent of customers receive the service for​ half-price?
​(b) If the automotive center does not want to give the discount to more than 2​% of its​ customers, how long should it make the guaranteed time​ limit?

Answer 1

• Calculate The Time Limit:

Using the Z-SCORE FORMULA, We can find the time limit (X) corresponding to the Z-SCORE of 2.05:

X = μ + Z * σ

= 17 + 2.05 * 2

= 21.1

• HENCE, ANSWER/OPTION (B): The Automotive Center should make the Guaranteed time Limit Approximately 21.1 Minutes to NOT GIVE the DISCOUNT to MORE than TWO-PERCENT (2%) of its customers.

• Step-by-step explanation:
• MAKE A PLAN:

Calculate the PERCENTAGE of CUSTOMERS RECEIVING HALF-PRICE SERVICE.

• Using the Z-SCORE FORMULA:

X = μ + Z * σ

• Since The Automotive service will take no longer than TWENTY (20) MINUTES, We need to find the PERCENTAGE of CUSTOMERS who received the service in more than TWENTY (20) Minutes.

1 - 0.9332 = 0.0668

• SOLVE THE PROBLEM: CONVERT THE PROBABILITY TO A PERCENTAGE:

0.0668 * 100 = 6.68%

• ANSWER (A):

APPROXIMATELY 6.68% of CUSTOMERS RECEIVED SERVICE for HALF-PRICE.

• (B) - STEP ONE (1): μ, σ,

Find the Z - SCORE for TWO - PERCENT (2%) PROBABILITY

• USING a Z-TABLE, We find the Z-SCORE corresponding to a probability of 0.98: SiNCE, we want to see the time limit for 98% of customers.):

Z = 2.05

• DRAW THE CONCLUSION: Calculate The Time Limit:

Using the Z-SCORE FORMULA, We can find the time limit (X) corresponding to the Z-SCORE of 2.05:

X = μ + Z * σ

= 17 + 2.05 * 2

= 21.1

• HENCE, ANSWER/OPTION (B): The Automotive Center should make the Guaranteed time Limit Approximately 21.1 Minutes to NOT GIVE the DISCOUNT to MORE than TWO-PERCENT (2%) of its customers.

I hope this helps you!