7. Hypothesis Testing: One-sample t-test
One-sample t-test: Purpose, Hypotheses, and Assumptions
One of the drawbacks of using a #Z#-test to test hypotheses about a population mean #\mu# is that it requires more information than is usually available.
Specifically, we need to know the value of the population standard deviation #\sigma# in order to calculate the #Z#-statistic:
\[Z = \cfrac{\bar{X} - \mu_0}{\sigma_{\bar{X}}} = \cfrac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}\]
This information is very rarely available in 'real-life', however, which drastically limits the practical applicability of the #Z#-test.
When we want to perform a hypothesis test for a population mean #\mu#, but the value of #\sigma# is unknown, we will have to use a one-sample Student's #t#-test instead.
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Student's t-test
A Student's #\boldsymbol{t}#-test is any statistical test for which the distribution of the test statistic follows a Student's #t#-distribution under the null hypothesis.
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One-sample t-test: Purpose and Hypotheses
A one-sample #\boldsymbol{t}#-test is used to test hypotheses about an unknown population mean #\mu#. It is used instead of the one-sample #Z#-test at times when the population standard deviation #\sigma# is unknown.
Here, one-sample indicates that a single sample is analyzed to draw inferences about a single population.
The hypotheses of a one-sample #t#-test for a population mean #\mu# are identical to the hypotheses of a one-sample #Z#-test for #\mu#:
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Assumptions of the One-Sample t-test
The following assumptions are required to hold in order for a one-sample t-test to produce valid results:
- Random sampling is used to draw the samples.
- Independence of observations, meaning the occurrence of one observation does not influence the chances of another observation occurring.
- The sampling distribution of the sample mean is approximately normal. The one-sample #t#-test is quite robust to violations of normality, meaning that the assumption of normality can be slightly violated and the test will still produce valid results.
