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

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

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.