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a) 22.66% of customers receive the service for half-price.

b) The guaranteed time limit should be of 25.2 minutes.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

(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?

Longer than 20 minutes is 1 subtracted by the pvalue of Z when X = 20. So

has a pvalue of 0.7734

1 - 0.7734 = 0.2266

22.66% 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?

The time limit should be the 100 - 2 = 98th percentile, which is X when Z has a pvalue of 0.98. So X when Z = 2.054.

The guaranteed time limit should be of 25.2 minutes.