Till now, we discussed about the one sample z test,t test and p test.

We are going to discuss about the two sample z test, p test, t test and paired t test.
Before talking on two sample z test, let us understand the difference between one sample z test and two sample z test.

One sample z test vs Two sample z test :

One sample z test- Let’s say there is a population p which has µ (mean) and standard deviation(σ). We select the sample (q) from p ,which has x(mean) and standard deviation(s). Using one sample z test ,we can conclude from sample whether population mean is changed or not.

Two sample z test- Lets say there are two population and find out whether two population has same mean or different using samples. Using two sample z test, we derive that there is a significant difference of mean for two population or not.

Example 1- There are two cities and if want to compare the mean salaries of two population in the city from the samples whether there is some significant difference between the population mean in the salary or not.

Example-2-Two machines producing some X product and want to compare whether both of the machine has same mean to produce the product or there is significant difference of means between the machines from the samples.

Two sample z test-

Image 1 and 2 -Two sample Z test

One may be confused , why two formula for two sample z test. It is even valid and good question.

Answer is that we can derive the first formula from second formula. Let’s see how

delta= µ1 – µ2

Here, delta is the difference of population means and it is 0 in case need to find whether there is difference of population means or not. So we put delta as 0, it forms the first equation.

Question- There are two machines A and B in the factory and 100 samples are drawn from each machine . Sample mean and standard deviation are 150 and 2 respectively for Machine A and sample mean and standard deviation are 152 and 2.5 respectively for Machine B. We need to find whether there is a significant difference between means of Machine A and B or not with 95% of confidence level ?

Answer :Null Hypothesis:µ1 – µ2=0
Alternate Hypothesis: µ1 ≠µ2

Here delta is 0 so we will use first formula of two sample z test. Numerator will be (152-150) and n1=100,n2=100,σ1=2 and σ2=2.5 , when we put the these values in the formula , denominator will be approx 0.3

z = (152-150)/0.3 = 6.6

z crtical value for 95% confidence level= 1.96

z > z critical so we can reject the null hypothesis ,means alternate hypothesis is correct so there is a change in the population means of two machines.


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