The authors generac generator produces voltage amounts with a mean of 125.6 volts and a standard deviation of 0.3 volt. using the Chebyshev's theorem, what do we know about the percentage of voltage amounts that are within 3 standard deviations of the mean? what are the minimum and maximum voltage amounts that are within 3 standard deviation of the mean?

Question

The authors generac generator produces voltage amounts with a mean of 125.6 volts and a standard deviation of 0.3 volt. using the Chebyshev's theorem, what do we know about the percentage of voltage amounts that are within 3 standard deviations of the mean? what are the minimum and maximum voltage amounts that are within 3 standard deviation of the mean?

Answer 1

This theorem says:

1. Within two standard deviations of the mean, you will find at least 75% of the data.
2. Within three standard deviations of the mean, you will find at least 89% of the data.

Here's how the formula shows this:

Formula is 1 - (1/k^2) ---> ^2 means squared.

If k = 2 (representing two standard deviations), we have this:
1 - (1/2^2) = 1 - (1/4) = 3/4 or .75 or 75%

If k = 3 (representing three standard deviations), we have this:
1 - (1/3^2) = 1 - (1/9) = 8/9 or approximately .89 or 89%

I'll let you take it from here.

Answer 2

To answer this question using Chebyshev's theorem, we need to apply the following inequality:

P(|X - μ| ≥ kσ) ≤ 1/k^2

where P is the probability, X is the random variable (voltage amount in this case), μ is the mean, σ is the standard deviation, and k is the number of standard deviations from the mean.

In this case, we want to find the probability that the voltage amounts are within 3 standard deviations of the mean. Therefore, k = 3.

Using Chebyshev's theorem, we can determine that the percentage of voltage amounts within 3 standard deviations of the mean is at least:

1 - 1/k^2 = 1 - 1/3^2 = 1 - 1/9 = 8/9 = 0.8889 (or approximately 88.89%)

So, we know that at least 88.89% of the voltage amounts will fall within 3 standard deviations of the mean.

To find the minimum and maximum voltage amounts within 3 standard deviations of the mean, we can calculate as follows:

Minimum voltage amount = μ - k * σ
Maximum voltage amount = μ + k * σ

Plugging in the values:
Minimum voltage amount = 125.6 - 3 * 0.3 = 125.6 - 0.9 = 124.7 volts
Maximum voltage amount = 125.6 + 3 * 0.3 = 125.6 + 0.9 = 126.5 volts

Therefore, the minimum voltage amount within 3 standard deviations of the mean is 124.7 volts, and the maximum voltage amount is 126.5 volts.