Q. 3.118 - Elementary Statistics

Question

A quantitative data set of size $200$ has mean $20$ and standard deviation $4$. At least how many observations lie between $8$ and $32$ $?$

Step-by-Step Solution

Answer

At least $178$ observations lie between $8$ and $32$.

Step1: Given information.

We have been given that,

Mean=$20$, Standard deviation =$4$, data set= $200$

We need find the number of observations lie between $8$ and $32$.

 

Step2: Simplify.

Chebyshev's rule: At least $100\left(1-\frac{1}{k^2}\right)\%$ of the data values lie within $k$ standard deviation from the mean$\left(k>1\right)$.

Using Chebyshev's Rule with $k=3$ , we know that at least

  $100\left(1-\frac{1}{k^2}\right)\%=100\left(1-\frac{1}{3^2}\right)\%=100\left(\frac{8}{9}\right)\%=89\%$

is within three standard deviation from the mean.

Determine the value that are three standard deviation from the mean,

                $$\begin{gathered} 20-3\times 4=8 \\ 20+3\times 4=32 \end{gathered}$$

At least $89\%$ of the observations lie between $8$ and $32$.

The number of observation is the product of the percentage and sample size 

              $89\%\times 200=178$

Thus , at least $178$ observations lie between $8$ and $32$.