Q. 3.116 - Elementary Statistics

Question

A quantitative data set has size $80$ has mean $30$ and standard deviation $5$. At least how many observations lie between $20$ and $40$ $?$

Step-by-Step Solution

Answer

At least $60$ observations lie between $20$ and $40$.

Step1: Given information

We have been given that,

Mean= $30$, Standard deviation = $5$ , data size=$80$ 

We need to find the number of observations lie between the two given values. 

Step2: Simplify.

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

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

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

is within two standard deviation from the mean.

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

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

                                      $$\begin{gathered} 30-2\times 5=20 \\ 30+2\times 5=40 \end{gathered}$$

At least $75\%$ of the observations lie within $20$ and $40$.

The number of observations lie within two standard deviation of the mean is the product of the percentage and the sample size,

                       $75\%\times 80=60$

Thus , at least $60$ observations lie between $20$ and $40$.