# A step by step guide to population comparison

Population, statistics and …

What is comparison of two population ?

We Compare two population means from independent population to obtain their difference for hypothesis testing . It plays a great role to decide whether null hypothesis should be rejected or accepted.

Why is it so important ?

Let’s take a real world Scenario. Suppose you are a ration shop owner and there is certain fluctuation happened in sugar price and because of that you have sold sugar at two different prices i.e:- at 34/- and 45/- and next week you have sold the sugar without fluctuation in it’s cost ie:- at 44/- and you want to decide when you have been profited ?

when you have sold sugar at a constant cost or when you have fluctuated the sugar cost on the basis of market need.

How we will decide H0 and H1 in this scenario ?

We will decide null or alternative hypothesis on the basis of mean difference. Let me clear this by doing mathematical analysis.

## case 1 :

when difference between two population mean is zero.

when 𝜇1 =𝜇2

H0 = 𝜇1 -μ2 =0

𝐻1 = 𝜇1-μ2 !=0 (Two tailed hypothesis testing)

## case 2 :

Difference between two population mean is less than zero

when 𝜇1< =𝜇2

𝐻0H0 = 𝜇1-𝜇2 <=0

𝐻1H1 = 𝜇1-𝜇2 >0 (one tailed hypothesis testing)

## case 3 :

Difference between two population mean is less than D.

when 𝜇1< =𝜇2+D

H0 = 𝜇1 -𝜇2< =D

𝐻1 = 𝜇1 -𝜇2 >D(one tailed hypothesis testing)

Note:- when we are solving this kind of problem we need to focus on size of population if size < 30 then we will apply t-statistics else we will go with z-statistics.

Z-stats formulation for comparison of two population

Here x1 and x2 are mean value of population and S.E is standard error . S.E is formulated below as;

let’s make it easy by solving an example;

Problem Statement :Is there evidence to conclude that the number of people preferring Duracell battery is different from the number of people preferring Energizer battery, given the following: Population 1: Duracell n1 = 100 , x1 = 308 ,s1 = 84 & Population 2: Energizer n2 = 100,x2 = 254, s2 = 67 ?

t-stats formulation for comparison of two population

To apply t-stats we must are to sure that the size of population’s sample < 30.

Now main difference occurs between z-stats and t-stats when we calculate standard error.

Problem Statement :The manufacturers of compact disk players want to test whether a small price reduction is enough to increase sales of their product. Is there evidence that the small price reduction is enough to increase sales of compact disk players? Population 1: Before reduction n1 = 15 ,x1 = Rs. 6598 s1 = Rs. 844& Population 2: After reduction n2 = 12,x2 = RS. 6870,s2 = Rs. 669 ?

What is population proportion ?

It is defined as fraction value associated with the given population. In most common way it defines percentage value associated with given population.It is denoted by “p”. It is also used to analyse the hypothesis of dataset. Before diving deep into it we need to understand two approach.

Approach 1 :

p1 = x1/n1 and p2 = x2/n2 then ;

let’s understand above formulation with the help of an example;

Problem Statement : Comparisons of two population proportions when the hypothesized difference is zero Carry out a two-tailed test of the equality of banks’ share of the car loan market in 1980 and 1995. Population 1: 1980 ,n1 = 100,x1 = 53,𝑝 1 = 0.53 Population 2: 1985 , n2 = 100 ,x2 = 43,𝑝 2= 0.43 ?

Approach 2 :

p1 !=x1/n1 and p2!=x2/n2 then ;

Problem Statement : Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at least 2500(dollar) in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such buyers when no sweepstakes are on. Population 1: With sweepstakes , n1 = 300 , x1 = 120 , 𝑝 = 0.40 Population 2: No sweepstakes n2 = 700 , x2 = 140 , 𝑝 2= 0.20 ?