Use Chebyshev’s theorem to find what percent of the values will fall between 162 and 292 for a data set with a mean of 227 and standard deviation of 13.
That is within 5 standard deviations or "sigmas" of the mean.
99.9997% are in that range, according to normal distribution tables. I am not familiar with your theorem
To use Chebyshev's theorem to find the percent of values that will fall between a given range, you can use the following formula:
Percent = 1 - (1 / k^2)
Where k is the number of standard deviations away from the mean that defines the range.
In this case, we are given a mean of 227 and a standard deviation of 13. We want to find the percent of values that fall between 162 and 292.
First, we need to find the number of standard deviations away from the mean that corresponds to the lower and upper bounds of the range.
For the lower bound (162):
z = (x - μ) / σ
z = (162 - 227) / 13
z ≈ -5.00
For the upper bound (292):
z = (x - μ) / σ
z = (292 - 227) / 13
z ≈ 5.00
Now that we have the number of standard deviations (k) for the range (-5.00 to 5.00), we can calculate the percent using Chebyshev's theorem.
Percent = 1 - (1 / k^2)
Percent = 1 - (1 / 5^2)
Percent = 1 - (1 / 25)
Percent ≈ 0.96
Therefore, approximately 96% of the values will fall between 162 and 292 for this data set.