A taxicab company wanted to determine the fuel cost of its fleet. A sample of 30 vehicles was selected, an the fuel cost for the last mooth was tabulated for each vehicle. Later it was discovered that the highest amount was mistakenly recorded with an extra zero, so it was 10 times to actual amount. when the correction was made, this was still the highest amount. which of the following must have remained the same after the correction was made?

Options

A. Mean
B. Median
C. Mode
D. Range

Explanation

To solve the problem, we need to analyze the impact of correcting the mistakenly recorded highest fuel cost on various statistical measures: mean, median, mode, and range.

### Step-by-step Solution:

1. **Understanding the Data Set**:
- We have a sample of 30 vehicles with their fuel costs recorded as \( X_1, X_2, \ldots, X_{30} \).
- It is given that \( X_{30} \) (the highest recorded fuel cost) was mistakenly recorded as 10 times its actual value.

2. **Correcting the Mistake**:
- The actual value of \( X_{30} \) is \( \frac{X_{30}}{10} \).
- After correction, the new set of fuel costs is \( X_1, X_2, \ldots, X_{29}, \frac{X_{30}}{10} \).

3. **Analyzing the Mean**:
- The mean (average) is calculated by summing all values and dividing by the number of values.
- Before correction:
\[ \text{Mean}_{\text{before}} = \frac{X_1 + X_2 + \ldots + X_{30}}{30} \]
- After correction:
\[ \text{Mean}_{\text{after}} = \frac{X_1 + X_2 + \ldots + X_{29} + \frac{X_{30}}{10}}{30} \]
- The mean will decrease after the correction since \( \frac{X_{30}}{10} < X_{30} \).

4. **Analyzing the Median**:
- The median is the middle value when the data is arranged in order.
- Since \( X_{30} \) was the highest value and after correction \( \frac{X_{30}}{10} \) is still the highest, the median remains unchanged.
- The median is the same because the highest value does not affect the middle value of the ordered list.

5. **Analyzing the Mode**:
- The mode is the value that appears most frequently.
- If \( X_{30} \) was unique (not equal to any other value), the mode may change after correction. Thus, we cannot guarantee that the mode remains the same.

6. **Analyzing the Range**:
- The range is calculated as the difference between the highest and lowest values.
- Before correction, the range was \( X_{30} - X_1 \).
- After correction, the range becomes \( \frac{X_{30}}{10} - X_1 \), which will definitely change since \( \frac{X_{30}}{10} < X_{30} \).

### Conclusion:
After analyzing all the statistical measures, we conclude that the only measure that remains unchanged after the correction is the **median**.

### Final Answer:
The median must have remained the same after the correction was made.

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