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* the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean | * the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean | ||

* this is exactly the correction we need to account for poorly estimated [[Standard Error]] when the sample size is not big | * this is exactly the correction we need to account for poorly estimated [[Standard Error]] when the sample size is not big | ||

+ | |||

+ | == Alternatives to $t$-Tests == | ||

+ | * http://stats.stackexchange.com/questions/183456/have-the-reports-of-the-death-of-the-t-test-been-greatly-exaggerated | ||

$t$-tests is a family of Statistical tests that use $t$-statistics

- critical values come from the $t$-distribution - used for calculating $p$-values

The following tests are $t$-tests:

- One-Sample $t$-test - for comparing the mean of a sample against some given mean
- Two-Sample $t$-test - for comparing the means of two samples
- Paired $t$-test - for Matching Pairs setup
- Pairwise $t$-test - for comparing the means of more than two samples

Assumptions for $t$ tests are similar to the assumptions of the $z$-tests

- Observations are independent (if less than 10% of population is sampled, then we can make sure it's satisfied)
- Sample size is sufficiently large so C.L.T. holds
- Moderate skew, few outliers (not too extreme)

Sample Size

- the sample size can be smaller than for $z$-tests
- so it can be smaller than 30 - after 30 we can safely use $z$-tests with almost the same outcomes

$t$-distribution:

- the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean
- this is exactly the correction we need to account for poorly estimated Standard Error when the sample size is not big