Question

Apply​ Chebyshev's rule to answer the following question. A quantitative data set has mean 22 and standard deviation 3. At least what percentage of the observations lie between 16 and 28​?

Answer 1

Answer:

At least 75% of the observations lie between 16 and 28​.

Step-by-step explanation:

The Chebyshev’s Theorem says:

For any numerical data set, at least $1-\frac{1}{k^2}$ of the data lie within k standard deviations of the mean, that is, in the interval with endpoints $\mu \pm k\sigma$ for populations, where k is any positive whole number that is greater than 1.

Given:

$\mu=22\\\sigma=3$

The interval (16, 28) is the one that is formed by adding and subtracting two standard deviations from the mean.

$(\mu - k\sigma,\mu +k\sigma)\\(22 - 2\cdot3,22 + 2\cdot3)\\(16,28)$

For k = 2, we see that $1-\frac{1}{2^2}=1-\frac{1}{4}=\frac{3}{4}$, which is 75% of the data must always be within two standard deviations of the mean.

At least 75% of the observations lie between 16 and 28​.