Q. 3.121 - Elementary Statistics

Question

The data set has mean $25$ and standard deviation$5$ . Fill in the following blanks:
a. Approximately $68\%$ of the observations lie between_ and _
b. Approximately  $95\%$of the observations lie between  _and _
c. Approximately $99.7\%$ of the observations lie between _ and _

Step-by-Step Solution

Answer

a. Approximately $68\%$of the observations lie between$20$ and $30$
b. Approximately $95\%$ of the observations lie between $15$ and $35$
c. Approximately $99.7\%$ of the observations lie between $10$and $40$

Part (a) Step 1: Given information

We are given that  data set has mean $25$ and standard deviation$5$  .

Part (a) Step 2: Explanation

Here we will use Empirical Rule, according to it , approximately , $68\%$of observations are within $1$standard deviation of mean .

So we will use formula  $\mu -\sigma =25-5=20$, here  is mean which is $25$  and $\sigma =5$ which is standard deviation.

Similarly , $\mu +\sigma =25+5=30$

So answer is that it is between$20$ and$30$

Part (b) Step 1: Given information

We are given that  data set has mean $25$ and standard deviation$5$. 

Part (b) Step 2: Explanation

Here we will use Empirical Rule, according to it , approximately , $95\%$  of observations are within$2$standard deviation of mean .

So we will use formula , $\mu -2\sigma =25-10=15$ , here  is mean  and $\sigma =5$ which is standard deviation .

Similarly , $\mu +2\sigma =25+10=35$

 So answer is that it is between$15$  and $35$

Part (c) Step 1: Given information

We are given that  data set has mean $25$ and standard deviation$5$ .

Part (c) Step 2: Explanation

Here we will use Empirical Rule, according to it , approximately ,$99.7\%$   of observations are within $3$ standard deviation of mean .

So we will use formula , $\mu -3\sigma =25-15=10$ , here $\mu =25$ is mean   and$\sigma =5$  which is standard deviation .

Similarly ,$\mu +3\sigma =25+15=40$ 

So answer is that it is between $10$ and $40$