Q. 11 - Elementary Statistics

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 $33$ ?

Step-by-Step Solution

Answer

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

Step 1: Given information

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

Step 2: 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 $144$