Q. 10 - Elementary Statistics

Question

A data set with roughly a bell-shaped distribution has mean $45$and standard deviation $12.$ Approximately what percentage of the observations lie between $33$and $57$? 

Step-by-Step Solution

Answer

The percentage of the observations lie between$33$and $57$approximately are $68\%$

Step 1: Given information

We are given that a data set with roughly a bell-shaped distribution has mean $45$and standard deviation $12$ .

Step 2: Explanation

We will use empirical rule,  to determine approximate values for the  of observations between $33$and $57$.

We are given that $\mu =45$ and$\sigma =12$ , where $\mu$is mean and $\sigma$ is  standard deviation.

Also , the given  interval $\left(33,57\right)$ 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(33,57\right)$$\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 $68\%$