Question

choose the answer that best describes the chebyshevs rule and fill in the blank with the appropriate percentage value.

a. chebyshevs rule says that approximately % of the data values lie within three standard deviations to either side of the mean.
b. chebyshevs rule says that at least % of the data values lie within three standard deviations to either side of the mean.
c. chebyshevs rule says that exactly % of the data values lie within three standard deviations to either side of the mean.

Explanation:

Step1: Recall Chebyshev's rule formula

Chebyshev's rule states that for any number \(k> 1\), the proportion of data within \(k\) standard - deviations of the mean is at least \(1-\frac{1}{k^{2}}\).

Step2: Substitute \(k = 3\) into the formula

When \(k = 3\), we calculate \(1-\frac{1}{k^{2}}=1-\frac{1}{9}=\frac{8}{9}\approx0.889 = 88.9\%\). Chebyshev's rule gives a lower - bound, so it says that at least \(88.9\%\) (or about \(89\%\)) of the data values lie within three standard deviations to either side of the mean.

Answer:

B. Chebyshev's rule says that at least 88.9% of the data values lie within three standard deviations to either side of the mean.