Test Statistic

A test statistic is a statistic (a quantity derived from the sample) used in Hypothesis Testing.

The [[Random Variable|r.v.]] $D$ is a test statistic if:  
1. The distribution of $D$ is known if $H_0$ is true.  
2. When $D=0$, we have the strongest possible evidence in  
favour of $H_0$, $D \geq 0$, and the larger $|D|$ becomes the more  
evidence we see against $H_0$.
 
Note*: $D$ is a discrepancy measure -> how much your data disagrees with $H_0$

There are so many test statistics, and the confidence interval and everything, it’s honestly getting a little overwhelming??

I also don’t think the table below is useful, I might as well just separate these into chunks.

Testing of Two means

\hline \text{Test Statistic} & Notes \\ \hline \frac{\overline{y}_d}{\frac{s_d}{\sqrt n}} & \text{Matched Pairs; d.f.} = n-1\\ [9pt] \hline \frac{a}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} & {\text{Unmatched pairs, large sample; d.f.}= n_1 + n_2 -2}\\ [9pt] \hline F & \text{Unmatched pairs, equal variance}\\ [9pt] \hline \end{array}$$ Other Tests $$\begin{array}{|c|c|} \hline \text{Test Statistic} & \text{Notes} \\ \hline \frac{\widehat{\beta}- \beta_0}{s_e / \sqrt{S_{xx}}}\\ \hline \lambda = 2 \sum_{j=1}^n y_j \ln \frac{y_j}{e_j}& \text{Goodness of Fit;} \chi^2(k-1-p) \\ \hline \lambda = 2 \sum_{i=1}^a \sum_{j=1}^b & \text{Independence of Attributes;} \chi^2(a-1)(b-1)\\ \hline \end{array}$$ Z-table if you know population mean and variance. When you don't, use t-table. I need to be able to understand when to use each test, [[notes/Student's t-Distribution|t-table]] or [[notes/Z-Score|Z-Table]] or [[notes/Chi-Squared Distribution|Chi-Squared Table]]. Matched pairs would be sort of this idea that we match two individuals together, there is no matching going on. You have unmatched Can you assume that the variances are equal or not? testing of two means