[Statistic (Summary)] T-test & $\chi^2$-test

T Test

One sample t-value:

\begin{equation} \frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}} \end{equation}

df = n-1

Two sample t-value

(assume equal population variance & std)
\begin{equation} \frac{\bar{x_1}-\bar{x_2}}{\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}} \end{equation}

where $s_p^2$ :
\begin{equation} \frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n1+n2-2} \end{equation}
df = $n_1$ + $n_2$ -2

(assume unequal population variance & std)
\begin{equation} \frac{\bar{x_1}-\bar{x_2}}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} \end{equation}
df = $n_1$-1 & $n_2$-1 중 smaller one

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Chisquare Test

One sample $\chi^2$ value :

\begin{equation*} \frac{\sum{(observed_i-expected_i)^2}}{expected_i} \end{equation*}

df = n - 1

Two sample $\chi^2$ value :

same equation as one above with different expected

\begin{equation*} expected_{i,j} = \frac{(row_i\ total)(column_j\ total)}{total\ observation} \end{equation*}

df = (# of rows) * (# of columns)