Q. 3.111 - Elementary Statistics

Question

A quantitative data set has mean $30$ and standard deviation $4$. At least what percentage of the observations lie between $10$ and $50$$?$ 

Step-by-Step Solution

Answer

$96\%$ of the observations lie between$10$ and $50$.  

Step1: Given information.

We have been given that,

Mean =$30$ ,Standard deviation= $4$

We need to find the percentage of observations lie between$10$ and $50$.

Step2: Simplify.

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

Let us determine the number of standard deviation that $10$ and $50$ are from the mean,

                           $$\begin{gathered} \frac{10-mean}{s\tan dard deviation} =\frac{10-30}{4}=-5 \\ \frac{50-mean}{s\tan dard deviation}=\frac{50-30}{4}= 5 \end{gathered}$$

Thus, $10$ and $50$ are $5$ standard deviation from the mean.

Using Chebyshev's Rule with $k=5$,we know that at least

        $100\left(1-\frac{1}{K^2}\right)\%=100\left(1-\frac{1}{5^2}\right)\%=100\left(1-\frac{1}{25}\right)\%=96\%$

is within $5$ standard deviation from the mean.