MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
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
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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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