MATLAB: An Introduction with Applications
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ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Analysis of Automotive Center Service Times
The time taken by an automotive center to complete an oil change service on an automobile approximately follows a normal distribution. The mean time for the service is 19 minutes, with a standard deviation of 2 minutes.
#### Problems:
1. **Service Guarantee Analysis**
- **Given:** The automotive center guarantees customers that the service will not take longer than 20 minutes.
- **Condition:** If it does take longer, the customer will receive the service for half-price.
- **Question:** What percent of customers receive the service for half-price?
2. **Customer Discount Limitation**
- **Given:** The automotive center aims to limit discounts to no more than 2% of its customers.
- **Question:** How long should the guaranteed time limit be to meet this criterion?
#### Solutions:
**(a) Percent of Customers Receiving Half-Price Service**
To determine the percentage of customers who receive the service for half-price, follow these steps:
1. Calculate the z-score for 20 minutes using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where \( X = 20 \) minutes, \( \mu = 19 \) minutes (mean), and \( \sigma = 2 \) minutes (standard deviation).
2. Substituting the values:
\[
z = \frac{20 - 19}{2} = 0.5
\]
3. Using a standard normal distribution table or a z-score calculator, find the area to the right of \( z = 0.5 \), which represents the percentage of customers receiving the half-price service.
- From the table, \( P(Z > 0.5) \approx 1 - 0.6915 = 0.3085 \).
Thus, \( 30.85\% \) of customers receive the service for half-price.
**(b) Time Limit for Not More Than 2% Discounts**
To ensure that no more than 2% of customers get the service for half-price:
1. Determine the z-score that corresponds to the top 2% of a normal distribution. This z-score is approximately \( z = 2.05 \).
2. Solve for \( X \) (the guaranteed time limit) using the z-score formula:
\[
X = z\sigma + \mu](https://content.bartleby.com/qna-images/question/9885b298-0da3-45d1-bdbe-d5991ebff07e/52664556-9fd8-436a-8182-69df349b5472/mh2p4b9_processed.png)
Transcribed Image Text:### Analysis of Automotive Center Service Times
The time taken by an automotive center to complete an oil change service on an automobile approximately follows a normal distribution. The mean time for the service is 19 minutes, with a standard deviation of 2 minutes.
#### Problems:
1. **Service Guarantee Analysis**
- **Given:** The automotive center guarantees customers that the service will not take longer than 20 minutes.
- **Condition:** If it does take longer, the customer will receive the service for half-price.
- **Question:** What percent of customers receive the service for half-price?
2. **Customer Discount Limitation**
- **Given:** The automotive center aims to limit discounts to no more than 2% of its customers.
- **Question:** How long should the guaranteed time limit be to meet this criterion?
#### Solutions:
**(a) Percent of Customers Receiving Half-Price Service**
To determine the percentage of customers who receive the service for half-price, follow these steps:
1. Calculate the z-score for 20 minutes using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where \( X = 20 \) minutes, \( \mu = 19 \) minutes (mean), and \( \sigma = 2 \) minutes (standard deviation).
2. Substituting the values:
\[
z = \frac{20 - 19}{2} = 0.5
\]
3. Using a standard normal distribution table or a z-score calculator, find the area to the right of \( z = 0.5 \), which represents the percentage of customers receiving the half-price service.
- From the table, \( P(Z > 0.5) \approx 1 - 0.6915 = 0.3085 \).
Thus, \( 30.85\% \) of customers receive the service for half-price.
**(b) Time Limit for Not More Than 2% Discounts**
To ensure that no more than 2% of customers get the service for half-price:
1. Determine the z-score that corresponds to the top 2% of a normal distribution. This z-score is approximately \( z = 2.05 \).
2. Solve for \( X \) (the guaranteed time limit) using the z-score formula:
\[
X = z\sigma + \mu
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