Q. 3.98 - Elementary Statistics

Question

Apply chebyshev's rule with $k=3$ to verify that at least $89\%$ of the observations in any data set lie within three standard deviations to either side of the mean, that is, between $\hat{x}-3s$ and $\hat{x}+3s$.

Step-by-Step Solution

Answer

According to Chebyshev's rule, for any quantitative data collection with $k=3$at least $89\%$ of the observations are within standard deviations of the mean, that is, $\hat{x}-3s$ and $\hat{x}+3s$. 

Step 1: Given information

We have been given that $k=3$.

We need to prove that at least $89\%$ of the observations in any data set lie within three standard deviations to either side of the mean, that is, between $\hat{x}-3s$ and $\hat{x}+3s$.

Step 2: Proof

Chebyshev's rule states that for any quantitative data collection with a real number $k$ higher than or equal to $1$, at least $1-\frac{1}{k^2}$ of the observations are within  standard deviations of the mean, that is, $\hat{x}-ks$ and $\hat{x}+ks$.

By Chebyshev's rule, for $k=3$

$1-\frac{1}{k^2}=1-\frac{1}{3^2}=1-\frac{1}{9}=\frac{8}{9}=88.89\%\simeq 89\%$

Weighing the weights of youngsters, for example, we discovered that our sample has a mean of $20$kg and a standard deviation of $5$kg. We know that at least $89\%$of the youngsters have weights that are three standard deviations off the mean because of chebyshev's criterion. We get $15$ by multiplying the standard deviation by $3$.Subtract and add this to the $20$kg mean. This indicates that $89\%$ of the children weigh between $5$ and $35$ kilograms.