## 1.9 T-value

T is a family of distributions that can be used (among other things) as a model for the distribution of population mean differences that are estimated from sample data. The thicker tails come from the fact that we are somewhat uncertain of population variance when we estimate it from a sample.

The t-value is the ratio of the difference between the sample mean and the population mean to the standard error of the sample mean. Mathematically, the t-value is calculated as follows: \[t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}\] where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

We can use the means and standard deviations from two samples to calculate an observed value of t: \[t_{\mathrm{obs}}=\frac{\left(\bar{x}_1-\bar{x}_2\right)}{s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\] where \(s_p\) is the pooled standard deviation: \[s_p = \sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}\] and \(n_1\) and \(n_2\) are the sample sizes of the two samples and \(s_1\) and \(s_2\) are the sample standard deviations of the two samples.

When you repeatedly take two samples and subtract the means, you get a sampling distribution that, other than having “thicker” tails, appears to be very normal. T-value on the t-distribution divides the distribution into regions (quantiles), it positions that value of t on the theoretical distribution of t values appropriate for the combined size of the two samples.