Question

b) If the sample size is 150, by Chebyshev's rule, at least how many measurements are in the interval within two SDV of the mean? Show your reason or work. c) If the sample size is 150 and we assume the data set is mound shaped, by empirical rule, approximately how many measurements is beyond the interval two SDV of the mean? Show your reason or work. That means, they are either belowor above

Answer 1

a)

the chebyshevs inequality is given as the following:

$P[\left | Y-\mu \right |\geq k\sigma]\leq 1/k^{2}$

here $Y = \overline{X}$ and the standard deviation for mean is given by $\sigma/\sqrt{n}$ while $k=2$

$P[\left |\overline{X}-\mu \right |\geq 2\sigma/\sqrt{150} ]\leq 1/4$

this also implies that $P[\left |\overline{X}-\mu \right |\leq 2\sigma/\sqrt{150} ]\geq 3/4$

which can be interpreted as atleast 75% of observations can be found within 2 standard deviation of mean.

b) by similar logic atmost 25% of population lies beyond 2 standard deviation of mean.