Q. 3.117 - Elementary Statistics

Question

A quantitative data set of size $60$ has mean $100$ and standard deviation $16$. At least how many observation lie between $68$ and $132$ $?$

Step-by-Step Solution

Answer

At least $45$ observations lie between $68$ and $132$.

Step1: Given information.

We have been given that,

Mean = $100$, Standard deviation = $16$, data size= $60$

We need to find the number of observations lie between $68$ and $132$.

Step2: Simplify.

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

Using Chebyshev's Rule with $k=2$, we know that at least

             $100\left(1-\frac{1}{k^2}\right)\%=100\left(1-\frac{1}{2^2}\right)\%=100\times \frac{3}{4}\%=75\%$

is within two standard deviation from the mean.

Determine the values that are two standard deviation from the mean,

                           $mean-2\left(s\tan dard deviation\right)$   

                    $$\begin{gathered} 100-2\times 16=68 \\ 100+2\times 16=132 \end{gathered}$$

Therefore , at least $75\%$of the observations lie between $68$ and $132$.

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

                 $75\%\times 60=45$

Thus, at least $45$ observations lie between $68$ and $132$.