2.5 Some statistical test

2.5.1 Parametric statistical test

2.5.1.1 $t$-test

2.5.1.1.1 One sample $t$-test

This test is used to determine if the mean of a single population is significantly different from a specified value $\mu_0$.

• • Null hypothesis: $H_0: \mu = \mu_0$
  • Alternative hypothesis: $H_1: \mu \neq \mu_0$ (two-sided), $H_1: \mu > \mu_0$ (one-sided), or $H_1: \mu < \mu_0$ (one-sided)

• $$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$ where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, and $n$ is the sample size.

• $df = n - 1$

• Reject $H_0$ if $|t| > t_{\alpha/2, df}$ (two-sided) or $t > t_{\alpha, df}$ (one-sided, upper tail) or $t < -t_{\alpha, df}$ (one-sided, lower tail), where $t_{\alpha/2, df}$ and $t_{\alpha, df}$ are the critical values from the $t$-distribution with $df$ degrees of freedom and significance level $\alpha$.

2.5.1.1.2 Two sample $t$-test

This test is used to determine if the means of two independent populations are significantly different, assuming that the populations have equal variances.

• • Null hypothesis: $H_0: \mu_1 = \mu_2$
  • Alternative hypothesis: $H_1: \mu_1 \neq \mu_2$ (two-sided), $H_1: \mu_1 > \mu_2$ (one-sided), or $H_1: \mu_1 < \mu_2$ (one-sided)

• $$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$ where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $n_1$ and $n_2$ are the sample sizes, and $s_p$ is the pooled sample standard deviation:

  $$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$ where $s_1$ and $s_2$ are the sample standard deviations.

• $df = n_1 + n_2 - 2$

• Reject $H_0$ if $|t| > t_{\alpha/2, df}$ (two-sided) or $t > t_{\alpha, df}$ (one-sided, upper tail) or $t < -t_{\alpha, df}$ (one-sided, lower tail).

2.5.1.1.3 Welch’s $t$-test

This test is used to determine if the means of two independent populations are significantly different, without assuming that the populations have equal variances.

• • Null hypothesis: $H_0: \mu_1 = \mu_2$
  • Alternative hypothesis: $H_1: \mu_1 \neq \mu_2$ (two-sided), $H_1: \mu_1 > \mu_2$ (one-sided), or $H_1: \mu_1 < \mu_2$ (one-sided)

• $$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$

• $$df \approx \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}$$

• Reject $H_0$ if $|t| > t_{\alpha/2, df}$ (two-sided) or $t > t_{\alpha, df}$ (one-sided, upper tail) or $t < -t_{\alpha, df}$ (one-sided, lower tail). Note that the degrees of freedom will likely be a non-integer value, so you’ll need to interpolate in a t-table or use statistical software.

2.5.1.1.4 Comparing Paired Samples

https://en.wikipedia.org/wiki/Paired_difference_test

2.5.1.2 $F$-test

2.5.1.3 $\chi^2$-test

2.5.1.3.1 for goodness-of-fit

2.5.1.3.2 for Homogeneity

2.5.1.3.3 for Independence

2.5.2 Non-parametric test

2.5.2.1 Mann–Whitney $U$-test (Wilcoxon rank-sum test)

2.5.2.2 Wilcoxon signed-rank test

2.5.2.3 Kolmogorov–Smirnov test

2.5.2.4 Lilliefors test