Q. 3.128 - Elementary Statistics

Question

The data set has $80$ observations and has mean $30$ and standard deviation $5$. Approximately how many observations lie between$20$ and$40$ ? 

Step-by-Step Solution

Answer

The observations that  lie between$20$ and$40$ are approximately $76$

Step 1: Given information

We are given that  data set has $80$  observations and has mean $30$ and standard deviation $5$.

Step 2: Explanation

We will use empirical rule,  to determine approximate values for the  of observations between $20$and $40$.

We are given that $\mu =30$ and$\sigma =5$ , where  is mean and  is  standard deviation.

Also , the given  interval $\left(20,40\right)$ 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  , $\left(\mu -k\sigma ,\mu +k\sigma \right)=(20,40)\Rightarrow k=2$

By empirical rule , $P\left(\mu -2\sigma <X<\mu +2\sigma \right)\approx 0.95$ where  is given interval.

So percentage is $95\%$

To find observations, we will use $95\%$of $80=76$