why-t-test-dont-work-with-large-samples

statistics

Usually, the goal of statistical tests is to show with high degree of confidence that an empirically estimated statistic is similar to a theoretically derived statistic
- The null hypothesis (the two statistics are similar) and alternative hypothesis (the two statistics are not) are presented to represent this where:
- $H_{null}: |\hat{\mu}-\mu| <=\Delta, H_{alt}: |\hat{\mu} - \mu| > \Delta$
- The $p_{value}$ $p_{v a l u e}$ from the test is what is the likelihood that we can observe an value larger than the test-statistic under the static distribution from chance
  - It is used to reject the null hypothesis, if $p_{value} < \alpha$
  - We establish a critical value $\alpha$ that is the threshold for which we can evaluate the $p_{value}$ with
  - ==The critical value should be established with the context of sample size==
As the sample size $N$ $N$ increases, the standard error gets smaller and the test statistic gets larger and $p_{value}$ $p_{v a l u e}$ gets smaller
- Then, the $\alpha$ needed to reject the null hypothesis needs to decrease because any differences no matter how small will be “significantly” different than the theoretical value
- At some point the critical value $\alpha$ needs to be incredibly small for huge samples
A way to evaluate large samples is to calculate the confidence intervals and see if that embraces the theoretically derived statistic
t-test is used to to measure the mean, z-test is to measure the mean when there is a large sample and the distribution is normal