Question

A data set of size $152$with roughly a bell-shaped distribution has mean $25$and standard deviation$4$$4$.

Approximately how many observations lie between$17$ and $33$ ?

Short answer

The observations that lie between$17$and $33$ approximately are$144$

Step-by-step solution

Given information

We are given that a data set of size $152$with roughly a bell-shaped distribution has mean$25$and standard deviation $4$.

Explanation

We will use empirical rule, to determine approximate values for the of observations between $17$and $33$.

We are given that $\mu =25$and$\sigma =4$, where $\mu$is mean and$\sigma$ is standard deviation.

Also , the given interval $\left(17,33\right)$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)=\left(17,33\right)$$\Rightarrow k=2$

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

So percentage is $95\%$

To find observations, we will use $95\%$of $152$is $144.4$

So approximate value is$144$