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 _

Short 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 $40$

Step-by-step solution

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 $40$