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Let's solve the problem step by step. The task is to calculate the **coefficient of variation (CV)** for two sets of waiting times—Single Line and Individual Lines. Then, you need to compare their variations.

### Formula for Coefficient of Variation (CV):
\[
CV = \frac{\sigma}{\mu} \times 100
\]
where:
- \(\sigma\) = Standard deviation of the dataset  
- \(\mu\) = Mean of the dataset  

---

### Step 1: Extract the data  
**Single Line Data:**  
388, 394, 400, 408, 426, 438, 443, 462, 462, 462  

**Individual Lines Data:**  
250, 322, 348, 373, 400, 463, 463, 509, 556, 599  

---

### Step 2: Calculate the Mean and Standard Deviation

#### **For Single Line:**
\[
\mu_{\text{Single}} = \frac{388 + 394 + 400 + \ldots + 462}{10} = 424.3
\]
\[
\sigma_{\text{Single}} = \sqrt{\frac{\sum{(x_i - \mu)^2}}{n}} = 26.73
\]

#### **For Individual Lines:**
\[
\mu_{\text{Individual}} = \frac{250 + 322 + 348 + \ldots + 599}{10} = 428.3
\]
\[
\sigma_{\text{Individual}} = 104.76
\]

---

### Step 3: Calculate the Coefficient of Variation

#### **Single Line:**
\[
CV_{\text{Single}} = \frac{26.73}{424.3} \times 100 = 6.3\%
\]

#### **Individual Lines:**
\[
CV_{\text{Individual}} = \frac{104.76}{428.3} \times 100 = 24.5\%
\]

---

### Step 4: Compare the Variation

Since \(CV_{\text{Individual}} = 24.5\%\) is much larger than \(CV_{\text{Single}} = 6.3\%\), the variation for individual lines is higher.

---

### Correct Answer:
**Option A:**  
*The variation of the individual lines is much larger than the variation of the single line because its CV is larger than the CV for the single line data set.*

---

Would you like further explanation or calculations?  

Here are 5 related questions you might find interesting:
1. What is the interpretation of the coefficient of variation in real-life scenarios?
2. How does the CV help compare datasets with different means?
3. What are the limitations of using the coefficient of variation?
4. How would removing outliers impact the CV for the individual lines?
5. Can the CV be negative, and if so, what does it mean?

**Tip:** The CV is most useful when comparing datasets with very different means, as it standardizes the variability relative to the mean.

A Providence Hospital experiment involves two different waiting line configurations for patients arriving for admission. The waiting times (in seconds) are recorded with a single line configuration that feeds four stations and another configuration with individual lines at the four stations. Find the coefficient of variation for each of the two samples, and then compare the variation.

Learn how to calculate the coefficient of variation for waiting times at a hospital. This problem involves comparing two configurations: a single line feeding multiple stations and individual lines at each station. Detailed solution includes mean, standard deviation, and coefficient of variation calculations. Suitable for students in Grades 11-12.

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