Question

The data set has $150$observations and has mean $35$and standard deviation $4$. Approximately how many observations lie between $31$and $39$?

Short answer

Zero observations lies between $31$and $39$.

Step-by-step solution

Given information

We are given that the data has $150$observations and

Mean $\overline{x}=35$

Standard deviation $s=4$

We have to find number of observations between $31$and $39$

Simplify

According to Chebyshev's Rule

For any quantitative data set and any real number $k$greater than or equal to $1$, at least $1-\frac{1}{k^2}$of the observations lie within $k$standard deviations to either side of the mean, that is, between $\overline{x}-k\left(s\right)$and $\overline{x}+k\left(s\right)$

Now, according to given data

$$\begin{gathered} \overline{x}-k\left(s\right)=31 \\ 35-k\left(4\right)=31 \\ k=1 \\ \overline{x}+k\left(s\right)=39 \\ 35+k\left(4\right)=39 \\ k=1 \end{gathered}$$

Therefore, number of observations between $31$and $39$ are $1-\frac{1}{k^2}=1-1=0$