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You would use the Student's t-test if your sample sizes are small and you do not know the population standard deviation.
To use the t-test the samples must be independent (sampling with replacement, etc...), the populations must be approximately normal, and the population variances can be assumed to be equal.
Example: a manufacturer is comparing a new wax to her old wax product. She compares the time of effectiveness for each type. The null hypothesis is that mean time of effectiveness is the same for both types, while the alternative hypothesis is that the means differ. The sample sizes are 5 cars for each wax type. Note that the sample size is below the recommended 30.
After selecting a confidence level, the manufacturer computes the critical value using the degrees of freedom. Then the t-test statistic is computed (using the sample means, the number in each sample, and the sample standard deviations.) As usual, she compares the test statistic to the critical value to make the decision.
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