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$$?$

Short answer

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

Step-by-step solution

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$32$.

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 $32$.