A survey found that cab drivers were earning an average of 0.25 dollars per mile. The standard deviation was $0.02. Using Chebyshev’s theorem, the minimum percentage of the data values that will fall between $0.20 and $0.30 is what percent? 

To solve this problem using Chebyshev’s theorem, we need to determine the minimum percentage of data values that fall within a specified range of the mean, given the standard deviation.

### Given Information:

- Mean ($ \mu $) = $0.25 per mile
- Standard deviation ($ \sigma $) = $0.02 per mile
- Range: $0.20 to $0.30

### Step 1: Determine the number of standard deviations ($ k $) the range is from the mean

The range given is from $0.20 to $0.30. We need to calculate how many standard deviations these values are away from the mean:

1. Calculate the distance from the mean to the lower bound ($0.20):

   \[
   \text{Distance to lower bound} = 0.25 - 0.20 = 0.05
   \]

2. Calculate the distance from the mean to the upper bound ($0.30):

   \[
   \text{Distance to upper bound} = 0.30 - 0.25 = 0.05
   \]

Since both distances are the same (0.05), we can proceed using either one.

Now, find $ k $ (the number of standard deviations):

\[
k = \frac{\text{Distance from the mean}}{\sigma} = \frac{0.05}{0.02}
\]

\[
k = 2.5
\]

### Step 2: Apply Chebyshev’s theorem

Chebyshev’s theorem states that at least $ 1 - \frac{1}{k^2} $ of the data falls within $ k $ standard deviations of the mean.

Plug in $ k = 2.5 $:

\[
1 - \frac{1}{k^2} = 1 - \frac{1}{(2.5)^2}
\]

\[
1 - \frac{1}{6.25} = 1 - 0.16 = 0.84
\]

\[
0.84 \times 100 = 84\%
\]

### Conclusion

Using Chebyshev’s theorem, we can conclude that at least **84%** of the cab drivers’ earnings per mile will fall between $0.20 and $0.30.

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Would you like more details or have any questions?

Here are some related questions to consider:

1. What if we wanted to calculate the percentage of earnings falling within $0.22 to $0.28 using Chebyshev's theorem?
2. How would the result change if the standard deviation was $0.03 instead of $0.02?
3. Can we use the Empirical Rule for this problem? Why or why not?
4. What does it mean if the actual percentage of cab drivers falling within this range is higher than Chebyshev’s estimate?
5. How can Chebyshev's theorem be applied in other real-world scenarios?

**Tip:** Remember, Chebyshev’s theorem applies to any data distribution, whether normal or not, making it a valuable tool for analyzing data with unknown distributions.

Learn how to apply Chebyshev’s theorem to determine the minimum percentage of cab drivers' earnings that fall between $0.20 and $0.30 per mile. Explore the calculation steps involving mean, standard deviation, and the theorem itself. Suitable for advanced high school students and beyond.

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