Question

A quantitative data set of size$150$has mean $35$ and standard deviation$4$. At least how many observations lie between $23$and $47$?

Short answer

At least that observations lie between $23$and $47$are $133$

Step-by-step solution

Given information

We are given that the data set of size$150$has mean $35$and standard deviation$4$

We need to find the observations that lie between $23$and$47$.

Explanation

We know that from Chebyshev's Theorem, in any of data set , the percentage of values that fall within k standard deviation from mean is at least $1-\frac{1}{k^2}$.Also , here $k>1$.

We are given that$\stackrel{-}{x}=35$and $s=4$, where $\stackrel{-}{x}$is mean and $s$is standard deviation.

Also , the given interval $(23,47)$will be formed by adding and subtracting three standard deviations from the mean.

So using the value of mean and standard deviation , we get $\left(\stackrel{-}{x}-ks,\stackrel{-}{x}+ks\right)=\left(23,47\right)$, so from here .

We also know from Chebyshev’s Theorem, it is given that at least $1-\frac{1}{k^2}=\frac{8}{9}$of the data are within this interval. Now also $\frac{8}{9}$of $150$is $133.3$that implies at least $133.3$observations are in the interval. We will not take a fractional observation, so we get at least$133$observations must lie inside the interval .