Q. 3.122 - Elementary Statistics

Question

The data set has mean$15$ and standard deviation $2$. Approximately what percentage of the observations lie between $13$ and $17$ ? 

Step-by-Step Solution

Answer

The percentage of the observations lie between $13$ and $17$ is$68\%$

Step 1:Given information

We are given that data set has mean$15$and standard deviation$2$ .

Step 2: Simplify

We will use empirical rule,  to determine approximate values for the percentage of observations between $13$and $17$.

We are given that $\mu =15$ and$\sigma =2$ , where $\mu$ is mean and $\sigma$ is  standard deviation.

Also , the given  interval $(13,17)$ will be  formed by adding and subtracting  standard deviations from the mean.We will check for different values of $k$.

So using the value of mean and standard deviation , we get  , $\left(\mu -k\sigma ,\mu +k\sigma \right)=(13,17)\Rightarrow k=1$

By empirical rule , $P\left(\mu -\sigma <X<\mu +\sigma \right)\approx 0.68$ where $X$ is given interval.

So percentage is $68\%$