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The two sample t-test or independent samples test is a method of testing the unknown population means of two groups and seeing whether they are equal or not.
The two sample t-test is generally used when the dataset values are independent and are randomly sampled from two populations without any biases and two independent groups have equal variances. If the variances don’t differ, you use a different estimate of the standard deviation.
Although, in cases where the sample size is small, one might not be able to test for normality. What you can rather do is assume the normality of your data or make use of a nonparametric test that doesn’t assume normality.
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For using the two sample t-test, we need to have two variables, one which defines the two groups and the other is the measurement of interest. Another thing we have is a hypothesis that implies that the means of the two populations are different. An example of this is given by jmp:
For each group we will need the average, sample size and the standard deviation like below:
As it is clear that the averages of the two groups are not the same, we find the difference between the averages which will depict the difference in the population means for the two groups.
To calculate the standard deviation, we will first calculate the variance. This will form a combined estimate for the standard deviation which will adjust according to the group size.
Now for the variance, we will just take a square root of 38.88
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Using the averages, standard deviations and samples sizes, we will calculate the test statistics:
For this, we will compare the test statistics to the theoretical value from the t-distribution. It goes like this:
Hence, the t-value with α = 0.05 and 21 degrees of freedom is 2.080.