Question

The data set has $250$observations and has mean $100$and standard deviation $16$. Approximately how many observations lie between $52$and $148$.

Short answer

$222$observations lie between $52$and$148$.

Step-by-step solution

Given information

We are given that the data has $250$and

Mean $\overline{x}=100$

Standard deviation $s=16$.

We have to find number of observations between $52$and $148$.

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

$$\begin{gathered} \overline{x}-k\left(s\right)=52 \\ 100-k\left(16\right)=52 \\ k=3 \\ \overline{x}+k\left(s\right)=148 \\ 100+k\left(16\right)=148 \\ k=3 \end{gathered}$$

Therefore, number of observations between $52$and $148$are

$$\begin{gathered} 1-\frac{1}{k^2}=1-\frac{1}{3^2}=0.889 \\ 0.889\times 250\approx 222 \end{gathered}$$