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$$?$

Short answer

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

Step-by-step solution

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.

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,

$mean-2\left(s\tan darddeviation\right)$

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

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 $40$.