t-test

One sample: compare $\bar{ξ}$ to known $m$ with $σ^{2}$ from the sample.

We are using the same data (from our sample) to estimate our variance. We estimated from the sample so this is a tricky situation.

It may seem strange that we know the true mean $m$ but not the true variance (where we could use the z-test).

Application of the t-test

This is relevant in Quantitative Genetics. If we have two pure (homozygote) lines and cross them into an F1 hybrid, we would know the mean from the parent generation that should be expected in the offspring, but there would be no genetic variance in the homozygote parent generation, and suddenly there will be lots in the offspring. So the t-test would be relevant here.

The formula

Similar to the z-score we calculate a $t$ value from this formula which follows a specific kind of normal distribution.

t = \frac{\bar{ξ} - m}{\sqrt{\frac{σ^{2}}{n}}}

Note that this depends on $n$ . If we have a very small value of $n$ our test will not be very useful. We have degrees of freedom (d.f.) of $n - 1$ . This affects which table we would use to see the significance level of our t-value.

Just like with the $z - t e s t$ , we determine if our value is in either of the 2.5% tails and see if our data is an outlier given the null hypothesis.

Two-sample t-test

This is the most common kind of t test. We have two samples (one from the treatment group ${\bar{ξ}}_{1}$ , one from the control ${\bar{ξ}}_{2}$ ). We want to know if there is a significant difference between the two, or if the difference is just due to sampling error.

We also have the variance of the sample averages $σ_{1}^{2}$ and $σ_{2}^{2}$ . Both samples have $n$ individuals (for this example).

Based on some assumptions and our null hypothesis, there are two things that should be true:

If the sample variances are both part of the true variance, their average should approximate the true sample distribution: $σ^{2} = \frac{1}{2} (σ_{1}^{2} + σ_{2}^{2})$
Based on the null hypothesis, we should expect:

t = \frac{{\bar{ξ}}_{1} - {\bar{ξ}}_{2}}{\sqrt{\frac{2 σ^{2}}{n}}}

With degrees of freedom $2 (n - 1)$ . We can look up if this value is significant. Remember, prosecutor's fallacy and also even if it is significant it might not be a very big difference.

This is where statistics starts and our class ends :)