Q. 3.196 - Elementary Statistics

Question

Identify the statistic that is used to estimate

(a) A population mean.

(b) A population standard deviation.

Step-by-Step Solution

Answer

(a) Population mean can be calculated using $\mu =\underset{}{(\sum }x )/N$

(b) Population standard deviation can be calculated using $\sigma ={\left(\right[\underset{}{\sum {(x-u)}^2}]/N)}^{1/2}$

Part(a) Step 1: Given information

We need to explain statistics used in population mean

Part(a) Step 2: Explanation

The population is essentially a collection of items in statistics. This can be in the form of numbers, objects, or anything else. As a result, the population mean is simply the average of this set of things. It's usually used to find the data set's center.

A formula for Population Mean($\mu$)is given by:

Population Mean = Sum of All the Items / Number of Items

$\mu =\sum _{}x /N$

Where,

$X$ denotes total number of observed values.  

$N$ denotes number of observations in the population.

Part(b) Step 1: Given information

We need to find the statistic used in estimating population standard deviation  

Part(b) Step 2: Explanation

The square root of the variance of a set of integers is the population standard deviation. It's used to calculate a confidence interval before making a decision.

Following is the procedure to calculate population standard deviation:

1. Calculate the mean.

2. For each number: Subtract the mean. Square the result.

3. Calculate the mean of the squared differences calculated in step $2$. Mean of these squared numbers is variance.

4. Take the square root variance to obtain the population standard deviation.

Formula is 

$\sigma ={\left(\right[\underset{}{\sum {(x-u)}^2}]/N)}^{1/2}$

$\sigma$denotes population standard deviation

$x$  is an individual value

$u$ is the average of the population

$N$ is the total number of the population