Data analysis Part 8: Two sample t test with unequal variances

Data analysis Part 8: Two sample t test with unequal variances (Welch’s t-test)

Two sample t test is done to decide if the population mean of two batches of samples are is equal or not. 

Population mean of two batches of samples are equal or not?

To perform two sample t test with unequal variances, first F test is carried out to find out whether the variances of the two samples are equal or not.

Let us take two sets of data as follows (Actual weight of paracetamol tablets that are marketed as 500mg)

Batch1	Batch2
499.643	500.240
499.475	501.055
500.459	500.120
499.692	500.680
497.946	500.478
500.280	500.157
500.072	499.961
500.054	500.833
499.619	501.052
500.761	499.996
499.251	499.264
500.808	499.221
500.224	500.384
500.083	500.999
500.747	499.507
498.547	500.690
501.282	499.731
499.672	499.934
499.086	497.957
499.600	499.131
500.828	499.257
501.772	502.237
499.615	498.184
500.491	500.381
499.105	501.112

Result of F test suggest that the variances of the two batch of samples are not equal. 

t-Test: Two-Sample Assuming Unequal Variances

	Batch 1	Batch 2
Mean	499.965	500.102
Variance	0.725	0.904
Observations	25	25
Hypothesized Mean Difference	0
df	47
t Stat	-0.540
P(T<=t) one-tail	0.296
t Critical one-tail	1.678
P(T<=t) two-tail	0.592
t Critical two-tail	2.012

Two tailed vs one tailed t test 

Here we are open to the two types of outcomes i.e. printed weight is less or more than the actual weight. Hence, two tailed t test is applicable. 

If we want to know whether the printer weight is more than the actual weight or not, in that scenario one tailed t test has to be used. For two tailed test α = 0.05. For one tailed test α = 0.025. 

Result and discussion

As per Fig.1 (Two sample two tailed t-test), it can be seen that test statistic (t Static) falls within the specified critical value (t critical).  

• Acceptance of Null hypothesis means wt. of paracetamol tablet (population mean) from two batches are equal. (Batch 1 = Batch 2).
  

• Rejection of Null hypothesis/ acceptance of alternative hypothesis means  wt. of paracetamol tablet (population mean) from two batches are different. (Batch 1 ≠ Batch 2). 
  

In this case as per Fig.1., Null hypothesis is accepted. Hence it is conclusively proved that there is no difference between the weight of batch 1 and batch 2 and mean weight of batch 1 and 2 matches with each other.

Population mean of batch 1 = batch 2 (Statistically proven without individual's bias)

Reference

Hyperlinked in the text.