Price of Electricity

The Energy Information Administration records the price of electricity in the United States each month. In July 2013, the average price of electricity was 11.43 cents per kilowatt-hour. Suppose the standard deviation is 1.70 cents per kilowatt-hour.

What can you determine about these data by using Chebyshev's Inequality with [tex][tex]$K=3$[/tex][/tex]?

At least [tex][tex]$88.89\%$[/tex][/tex] of the data fall between [tex][tex]$\square$[/tex][/tex] and [tex][tex]$\square$[/tex][/tex] cents per kilowatt-hour.

(Round answers to two decimal places as necessary.)



Answer :

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.