According to Chebyshev's theorem, what proportion of a distribution will be within k = 4 standard deviations of the mean? Show all work as to how to find this.
Chebyshev's theorem states that for any distribution, regardless of shape, at least (1-1/k^2) proportion of the data will fall within k standard deviations of the mean. In this case, k = 4 standard deviations.
To find the proportion of a distribution that will be within 4 standard deviations of the mean using Chebyshev's theorem, we need to calculate (1 - 1/k^2), where k = 4.
First, compute the value of k^2:
k^2 = 4^2 = 16
Now, plug in the value of k^2 into the formula:
1 - 1/16 = 15/16
Therefore, the proportion of the distribution that will be within 4 standard deviations of the mean, according to Chebyshev's theorem, is 15/16 or approximately 0.9375.
So, approximately 93.75% of the data will fall within 4 standard deviations of the mean.