Question

The data set has $80$observations and has mean $30$and standard deviation $5$. Approximately how many observations lie between$20$ and$40$ ?

Short answer

The observations that lie between$20$and$40$ are approximately$76$

Step-by-step solution

Given information

We are given that data set has$80$ observations and has mean $30$and standard deviation$5$.

Explanation

We will use empirical rule, to determine approximate values for the of observations between $20$and $40$.

We are given that $\mu =30$and$\sigma =5$, where is mean and is standard deviation.

Also , the given interval $\left(20,40\right)$will be formed by adding and subtracting standard deviations from the mean .We will check for different values of .

So using the value of mean and standard deviation , we get ,$\left(\mu -k\sigma ,\mu +k\sigma \right)=(20,40)\Rightarrow k=2$

By empirical rule , $P\left(\mu -2\sigma <X<\mu +2\sigma \right)\approx 0.95$where is given interval.

So percentage is $95\%$

To find observations, we will use$95\%$of$80=76$