Q. 9 - Elementary Statistics

Question

 A quantitative data set of size $87$ has mean$80$ and standard deviation $10$. At least how many observations lie between $60$ and $100$ ? 

Step-by-Step Solution

Answer

The observations that lie between $60$ and $100$ at least are $65$

Step 1: Given information

We are given that  quantitative data set of size$87$  has mean$80$ and standard deviation $10$.

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 .Also , here .

We are given that$\stackrel{-}{x}=80$ and $s=10$, where is mean and  is  standard deviation.

Also , the given  interval $\left(60,100\right)$ will be  formed by adding and subtracting three standard deviations from the mean.

So using the value of mean and standard deviation , we get $\left(\stackrel{-}{x}-ks,\stackrel{-}{x}+ks\right)=\left(60,100\right)$ , so from here$k=2$ .

We also know from Chebyshev’s Theorem, it is given that  at least $1-\frac{1}{k^2}=\frac{3}{4}$of the data are within this interval. Now also  $\frac{3}{4}$ of $87$ is  that $65.25$ implies at least $65.25$observations are in the interval. We  will not  take a fractional observation, so we get  at least$65$ observations must lie inside the interval.