Question

The data set has mean $30$and standard deviation$4$ . Approximately what percentage of the observations lie between $22$and $38$?

Short answer

The percentage of the observations lie between $22$and $38$ is$95\%$.

Step-by-step solution

Given information

We are given that data set has mean $30$and standard deviation$4$ .

Explanation

We will use empirical rule, to determine approximate values for the percentage of observations between $22$and $38$.

We are given that $\mu =30$and$\sigma =4$ , where is mean and is standard deviation.

Also , the given interval $\left(22,38\right)$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)=$$(22,38)\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$95\%$