Q. 3.110 - Elementary Statistics

Question

A quantitative data set has mean $15$ and standard deviation $2$. At least what percentage of the observations lie between $7$ and $23$ $?$

Step-by-Step Solution

Answer

$93.75\%$ of the observations lie between $7$ and $23$.

Step1: Given information.

We have been given that ,

Mean = $15$, Standard deviation = $2$

We need to find the percentage of the observations that lie between $7$ and $23$.

Step2: Simplify.

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

Let us determine the number of standard deviation that $7$ and $23$ are from the mean.

                         $$\begin{gathered} \frac{7-mean}{s\tan dard deviation}=\frac{7-15}{2}=-4 \\ \frac{23-mean}{s\tan dard deviation}=\frac{23-15}{2}=4 \end{gathered}$$

So $7$ and $23$ are $4$ standard deviation from the mean.

Using the Chebyshev's Rule with $k=4$ , we know that at least,

               $100\left(1-\frac{1}{4^2}\right)\%=100\left(1-\frac{1}{16}\right)\%= 93.75\%$

is within $4$ standard deviation from the mean.