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Welch's t-test

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What is Welch’s t-test?

Welch’s t-test also known as unequal variances t-test is used when you want to test whether the means of two population are equal. This test is generally applied when the there is a difference between the variations of two populations and also when their sample sizes are unequal. 

To understand when you will apply the Welch’s t-test, let’s take a look at some scenarios which require you to use Welch’s t-test for better results:

  • When it is assumed that the distribution is normal.
  • When the samples have different variances.
  • Sample sizes of the populations are different.

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Assumptions and preleminary tests for Welch’s t-test

The Welch t-test assumes the following characteristics about the data:

  • Independence of the observations. Each subject should belong to only one group.
  • No significant outliers in the two groups
  • Normality. the data for each group should be approximately normally distributed.

How to perform Welch’s t-test?

In this we will learn a step-by-step guide on how to perform Welch’s t-test:

Step 1: State then null and alternative hypothesis. 

Like any other statistical test, Welch’s t-test requires a null and an alternative hypothesis based on which the results will be concluded. 

H0: μ1 = μ2 (Null hypothesis stating the means of two populations are the same)
Ha: μ2 ≠ μ1 (alternative hypothesis that states that the means of two population are nor equal)

Hence, we will perform a Welch’s t-test to decide which one of those can be rejected.

Step 2: Calculate the means of two samples

To calculate the means, take the sum of samples and then divide the count by the total number of items in the sample. 

Do this for both the populations. 

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Step 3: calculate the standard deviation 

In this step, you need to calculate the standard deviation of both the population samples using the following formula:

Step 4: Calculate the t value

To calculate the t value, use the formula:

Where,

  • X1 is the mean of the first sample.
  • s1 is the standard deviation of the first sample.
  • N2 is the first sample size.
  • The same is calculated for the second sample.

Step 5: Calculate the degree of freedom

The degree of freedom (v) can be calculated using the below formula:

Where v1 is the degrees of freedom from the first sample and is given by the formula: 

v1 = N1 – 1

And v2 is the degrees of freedom from the second sample and is given by the formula: 

v2 = N2 – 1

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Step 6: Compare the calculated t value with the tabulated t value

The calculated t value (test statistic) will be either greater than or less than the tabulated t value (critical value).

Step 7: Conclusion

We will decide if the null hypothesis is rejected or accepted based on the following results:

  • If the calculated test statistic > critical value, reject the null hypothesis.
  • If the calculated test statistic < or within the critical value, accept the null hypothesis.

Example of Welch’s t-test

You can calculate the Welch’s t-test through excel or using a statistical programming language R.

In this example, we will see how to manually calculate Welch’s t-test.

We will take two samples of their respective populations and calculate their mean, sample sizes and standard deviance. 

Sample 1: 14, 15, 15, 15, 16, 18, 22, 23, 24, 25, 25

Sample 2: 10, 12, 14, 15, 18, 22, 24, 27, 31, 33, 34, 34, 34

Mean of sample 1 (x1– 19.27
Mean of sample 2 (x2) – 23.69
Standard deviation of sample 1 (s12) – 20.42
Standard deviation of sample 2 (s22– 83.23
Sample size of sample 1 (n1) – 11
Sample size of sample 2 (n2) – 13

To calculate test statistics, we will substitute these values in the t value formula:

 (19.27 – 23.69) / (√20.42/11 + 83.23/13) =  (-4.42) / 2.873  =  -1.538

To calculate the degrees of freedom (v) we substitute the above values in the formula: (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 – 1) ] + [ (s22 / n2)2 / (n2 – 1) ] }

(20.42/11 + 83.23/13)2 / { [ (20.42/11)2 / (11 – 1) ] + [ (83.23/13)2 / (13 – 1) ] } = 18.137

We will round this to the next nearest integer of 18.

Next, we will find the critical value in the tabulated t-distribution table corresponding to the significance level = 0.05 and degrees of freedom as 18.

The critical t value is 2.101.

Since the test statistic value is 1.538, it is less than the critical t value. Hence we will accept the null hypothesis and state that there is not sufficient evidence to say that the means of the two populations are significantly different.

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