Why does standard error show up in a hypothesis test instead of just using the sample standard deviation?
The sample standard deviation measures dispersion of individual observations. The standard error measures dispersion of the sample mean. A hypothesis test about a population mean asks whether the observed sample mean is unusually far from the hypothesized mean, so the test statistic scales the difference by the standard error.
If the sample has 49 observations and a sample standard deviation of 14, the standard error of the mean is 14 divided by 7, or 2. A sample mean that is 6 units above the hypothesized mean is therefore 3 standard errors away. That is the intuition behind the test statistic.
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