Q. 3.119 - Elementary Statistics

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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$47$.

Step 2: 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 .