Q. 3.123 - Elementary Statistics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given information

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

Step 2 : 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 $95\%$