To determine the guaranteed time for the auto service shop such that no more than 1% of customers have to wait beyond this time, we need to calculate the time threshold associated with the 99th percentile of the normal distribution of oil change times. Let's go through the steps:
1. Understanding the Normal Distribution:
- The time taken for an oil change follows a normal distribution with a mean ([tex]\(\mu\)[/tex]) of 17.8 minutes and a standard deviation ([tex]\(\sigma\)[/tex]) of 5.2 minutes.
- We need to find the time [tex]\(X\)[/tex] such that the probability of the time exceeding [tex]\(X\)[/tex] is 1%. In other words, [tex]\(99\%\)[/tex] of the customers' service times will be less than or equal to this time [tex]\(X\)[/tex].
2. Finding the Z-Score:
- The Z-score associated with the 99th percentile can be found using statistical tables or standard tools. A Z-score tells us how many standard deviations an element is from the mean.
- For a 99% upper limit (which corresponds to the 99th percentile), the Z-score is approximately [tex]\(2.326\)[/tex].
3. Transforming the Z-Score to the X Value:
- We use the Z-score formula to convert back from the Z-score to the actual value [tex]\(X\)[/tex]:
[tex]\[
X = \mu + Z \cdot \sigma
\][/tex]
- Plugging in the values:
[tex]\[
X = 17.8 + 2.326 \cdot 5.2
\][/tex]
4. Calculating the Guaranteed Time:
- First, calculate the product of the Z-score and the standard deviation:
[tex]\[
2.326 \cdot 5.2 = 12.0976
\][/tex]
- Adding this to the mean:
[tex]\[
X = 17.8 + 12.0976 = 29.8976
\][/tex]
5. Rounding to the Nearest Minute:
- The final step is to round this calculated time to the nearest minute:
[tex]\[
29.8976 \approx 30
\][/tex]
Therefore, the auto service shop should set the guaranteed time to 30 minutes to ensure that no more than 1% of customers will qualify for a free lubrication job due to extended wait times.
