Q. 3.120 - Elementary Statistics

Question

The data set has mean $10$ and standard deviation$3$ . 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$3$ and $13$
b. Approximately  $95\%$ of the observations lie between $4$ and $16$
c. Approximately $99.7\%$ of the observations lie between$1$ and $19$

Part (a) Step 1: Given information

We are given data set has mean $10$ and standard deviation$3$. 

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  $\mu -\sigma =10-7=3$, here $\mu$ is mean which is $10$  and $\sigma =3$ which is standard deviation .

Similarly , $\mu +\sigma =10+3=13$

So answer is that it is between$3$ and$13$

Part (b) Step 1: Given information

We are given that data set has mean $10$ and standard deviation$3$ . 

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 , $\mu -2\sigma =10-6=4$ , here$\mu$  is mean which is $10$ and $\sigma =3$ which is standard deviation .

Similarly , $\mu +2\sigma =10+6=16$

 So answer is that it is between$4$  and $16$

Part (c) Step 1: Given information

We are given that data set has mean $10$ and standard deviation $3$ .

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 , $\mu -3\sigma =10-9=1$ , here $\mu$ is mean which is$\sigma$  and  which is standard deviation .

Similarly ,$\mu +3\sigma =10+9=19$ 

So answer is that it is between $1$ and $19$