To analyze the price of electricity with the provided data, we can use Chebyshev's Inequality, which is applicable to any probability distribution. Chebyshev's Inequality states that for any number [tex]\(K > 1\)[/tex], at least [tex]\((1 - \frac{1}{K^2})\)[/tex] of the data values lie within [tex]\(K\)[/tex] standard deviations of the mean.
Given the following information:
- Mean price ([tex]\(\mu\)[/tex]) = 11.43 cents per kilowatt-hour
- Standard deviation ([tex]\(\sigma\)[/tex]) = 1.70 cents per kilowatt-hour
- [tex]\(K = 3\)[/tex]
First, we determine the bounds within which at least a certain percentage of data will fall. This can be calculated using the formula for Chebyshev's Inequality:
[tex]\[
\text{Lower bound} = \mu - K \sigma
\][/tex]
[tex]\[
\text{Upper bound} = \mu + K \sigma
\][/tex]
Step-by-Step Calculation:
1. Calculate the lower bound:
[tex]\[
\text{Lower bound} = 11.43 - 3 \times 1.70
\][/tex]
[tex]\[
\text{Lower bound} = 11.43 - 5.10
\][/tex]
[tex]\[
\text{Lower bound} = 6.33
\][/tex]
2. Calculate the upper bound:
[tex]\[
\text{Upper bound} = 11.43 + 3 \times 1.70
\][/tex]
[tex]\[
\text{Upper bound} = 11.43 + 5.10
\][/tex]
[tex]\[
\text{Upper bound} = 16.53
\][/tex]
3. Determine the percentage of data within these bounds:
[tex]\[
\text{Percentage} = \left(1 - \frac{1}{3^2}\right) \times 100\%
\][/tex]
[tex]\[
\text{Percentage} = \left(1 - \frac{1}{9}\right) \times 100\%
\][/tex]
[tex]\[
\text{Percentage} = \left(\frac{8}{9}\right) \times 100\%
\][/tex]
[tex]\[
\text{Percentage} \approx 88.89\%
\][/tex]
Therefore, using Chebyshev's Inequality with [tex]\(K=3\)[/tex], we can determine that at least [tex]\(88.89\%\)[/tex] of the data fall between [tex]\(6.33\)[/tex] and [tex]\(16.53\)[/tex] cents per kilowatt-hour, rounded to two decimal places.
