R Code can be found in here
\[ P(\mathcal{X}=k)\ = {n \choose k}p^k(1-p)^{n-k} \]
$P(\mathcal{X}=k)\ = {n \choose k}p^k(1-p)^{n-k}$\(P(\mathcal{X}=k)\ = \frac{1}{\sqrt{2\pi}\sigma}*e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)
$P(\mathcal{X}=k)\ = \frac{1}{\sqrt{2\pi}\sigma}*e^{-\frac{(x-\mu)^2}{2\sigma^2}}$\(H_0\): there’s no difference betwee sample mean and the true mean
\(H_1\): the sample mean is lower than the true mean
\(\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}\)
\(s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}\)
\(t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}\)
\(Reject ~ H_0 ~ if ~ t<t_{n-1,\alpha}\)
\(Fail ~ reject ~ H_0 ~ if ~ t>t_{n-1,\alpha}\)
\(CI : (-\infty,\bar{x} - t_{df,\alpha}*sd/\sqrt{n})\)
$H_0$: there's no difference betwee sample mean and the true mean
$H_1$: the sample mean is lower than the true mean
$\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}$
$s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}$
$t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}$
$Reject ~ H_0 ~ if ~ t<t_{n-1,\alpha}$
$Fail ~ reject ~ H_0 ~ if ~ t>t_{n-1,\alpha}$
$CI : (-\infty,\bar{x} - t_{df,\alpha}*sd/\sqrt{n})$\(H_0\): there’s no difference betwee sample mean and the true mean
\(H_1\): the sample mean is greater than the true mean
\(\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}\)
\(s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}\)
\(t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}\)
\(Reject ~ H_0 ~ if ~ t>t_{n-1,1-\alpha}\)
\(Fail ~ reject ~ H_0 ~ if ~ t<t_{n-1,1-\alpha}\)
\(CI : (\bar{x} + t_{df,1-\alpha}*sd/\sqrt{n},\infty)\)
$H_0$: there's no difference betwee sample mean and the true mean
$H_1$: the sample mean is greater than the true mean
$\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}$
$s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}$
$t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}$
$Reject ~ H_0 ~ if ~ t>t_{n-1,1-\alpha}$
$Fail ~ reject ~ H_0 ~ if ~ t<t_{n-1,1-\alpha}$
$CI : (\bar{x} + t_{df,1-\alpha}*sd/\sqrt{n},\infty)$\(H_0\): there’s no difference betwee sample mean and the true mean
\(H_1\): the sample mean is different from the true mean
\(\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}\)
\(s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}\)
\(t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}\)
\(Reject ~ H_0 ~ if ~ |t|<t_{n-1,1-\alpha/2}\)
\(Fail ~ reject ~ H_0 ~ if ~ |t|<t_{n-1,1 - \alpha/2}\)
\(CI : (\bar{x} - t_{df,1-\alpha/2}*sd/\sqrt{n} ,\bar{x} + t_{df,1-\alpha/2}*sd/\sqrt{n})\)
$H_0$: there's no difference betwee sample mean and the true mean
$H_1$: the sample mean is different from the true mean
$\bar{x} = \frac{\sum_{i=1}^n{x_i}}{n}$
$s_d = \sqrt{\sum_{i=1}^n{(x_i - \bar{x})^2}/(n-1)}$
$t = \frac{\bar{x} - 0}{s_d/\sqrt{n}}$
$Reject ~ H_0 ~ if ~ |t|<t_{n-1,1-\alpha/2}$
$Fail ~ reject ~ H_0 ~ if ~ |t|<t_{n-1,1 - \alpha/2}$
$CI : (\bar{x} - t_{df,1-\alpha/2}*sd/\sqrt{n} ,\bar{x} + t_{df,1-\alpha/2}*sd/\sqrt{n})$\(H_0\): there’s no difference betwee difference
\(H_1\): the difference is different
\(\bar{d} = \frac{\sum_{i=1}^n{d_i}}{n}\)
\(s_d = \sqrt{\sum_{i=1}^n{(d_i - \bar{d})^2}/(n-1)}\)
\(t = \frac{\bar{d} - 0}{s_d/\sqrt{n}}\)
\(Reject ~ H_0 ~ if ~ |t|<t_{n-1,1-\alpha/2}\)
\(Fail ~ reject ~ H_0 ~ if ~ |t|<t_{n-1,1 - \alpha/2}\)
\(CI : (\bar{d} - t_{df,1-\alpha/2}*sd/\sqrt{n} ,\bar{d} + t_{df,1-\alpha/2}*sd/\sqrt{n})\)
$H_0$: there's no difference betwee difference
$H_1$: the difference is different
$\bar{d} = \frac{\sum_{i=1}^n{d_i}}{n}$
$s_d = \sqrt{\sum_{i=1}^n{(d_i - \bar{d})^2}/(n-1)}$
$t = \frac{\bar{d} - 0}{s_d/\sqrt{n}}$
$Reject ~ H_0 ~ if ~ |t|<t_{n-1,1-\alpha/2}$
$Fail ~ reject ~ H_0 ~ if ~ |t|<t_{n-1,1 - \alpha/2}$
$CI : (\bar{x} - t_{df,1-\alpha/2}*sd/\sqrt{n} ,\bar{d} + t_{df,1-\alpha/2}*sd/\sqrt{n})$\(H_0\): the variances between group are equal(no difference)
\(H_1\): the variances between group are not equal(there’s difference)
\(s_{x_1} = \sqrt{\sum_{i=1}^{n_1}(x_i - \bar{x_1})^2/(n_1-1)}\)
\(s_{x_2} = \sqrt{\sum_{j=1}^{n_2}(x_i - \bar{x_2})^2/(n_2-1)}\)
\(F = s_1^2/s_2^2 \sim F_{n_1-1,n_2-1}\)
\(Reject ~ H_0 ~ if ~ F>F_{n_1-1,n_2-1,1-\alpha/2} ~ OR ~ F<F_{n_1-1,n_2-1,\alpha/2}\)
\(Fail ~ reject ~ H_0 ~ if ~ F_{n_1-1,n_2-1,\alpha/2}<F<F_{n_1-1,n_2-1,1-\alpha/2}\)
$H_0$: the variances between group are equal(no difference)
$H_1$: the variances between group are not equal(there's difference)
$s_{x_1} = \sqrt{\sum_{i=1}^{n_1}(x_i - \bar{x_1})^2/(n_1-1)}$
$s_{x_2} = \sqrt{\sum_{j=1}^{n_2}(x_i - \bar{x_2})^2/(n_2-1)}$
$F = s_1^2/s_2^2 \sim F_{n_1-1,n_2-1}$
$Reject ~ H_0 ~ if ~ F>F_{n_1-1,n_2-1,1-\alpha/2} ~ OR ~ F<F_{n_1-1,n_2-1,\alpha/2}$
$Fail ~ reject ~ H_0 ~ if ~ F_{n_1-1,n_2-1,\alpha/2}<F<F_{n_1-1,n_2-1,1-\alpha/2}$\(H_0\): the means between group are equal(no difference)
\(H_1\): the means between group are not equal(there’s difference)
\(s_{pool} = \frac{(n_1-1)s_1 + (n_2 -1)s_2}{n_1+n_2-2}\)
\(t = \frac{\bar{X_1} - \bar{X_2}}{s_{pool}\times\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\)
\(Reject ~ H_0 ~ if ~ |t|>t_{df,1-\alpha/2}\)
\(Fail ~ reject ~ H_0 ~ if ~ |t|<t_{df,1-\alpha/2}\)
\(CI = (\bar{X_1}-\bar{X_2} ~ - ~ t_{df,1-\alpha/2}s_{pool}/\sqrt{1/n_1+1/n_2},\bar{X_1}-\bar{X_2} ~ + ~ t_{df,1-\alpha/2}s_{pool}/\sqrt{1/n_1+1/n_2} )\)
$H_0$: the means between group are equal(no difference)
$H_1$: the means between group are not equal(there's difference)
$s_{pool} = \frac{(n_1-1)s_1 + (n_2 -1)s_2}{n_1+n_2-2}$
$t = \frac{\bar{X_1} - \bar{X_2}}{s_{pool}\times\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$
$Reject ~ H_0 ~ if ~ |t|>t_{df,1-\alpha/2}$
$Fail ~ reject ~ H_0 ~ if ~ |t|<t_{df,1-\alpha/2}$
$CI = (\bar{X_1}-\bar{X_2} ~ - ~ t_{df,1-\alpha/2}s_{pool}/\sqrt{1/n_1+1/n_2},\bar{X_1}-\bar{X_2} ~ + ~ t_{df,1-\alpha/2}s_{pool}/\sqrt{1/n_1+1/n_2} )$\(H_0\): the means between group are equal(no difference)
\(H_1\): the means between group are not equal(there’s difference)
\(\bar{X_1} - \bar{X_2} \sim N(\mu_1-\mu_2,\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2})\) if we know the population variance
\(t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}~\sim t_{d''}\)
\(d' = round(d'') = \frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})^2}{\frac{s_1^2}{n_1}^2/(n_1-1)+\frac{s_2^2}{n_2}^2/(n_2-1)}\)
\(Reject ~ H_0 ~ if ~ |t|>t_{df,1-\alpha/2}\)
\(Fail ~ reject ~ H_0 ~ if ~ |t|<t_{df,1-\alpha/2}\)
\(CI = (\bar{X_1}-\bar{X_2} ~ - ~ t_{df,1-\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}},\bar{X_1}-\bar{X_2} ~ + ~ t_{df,1-\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}})\)
$H_0$: the means between group are equal(no difference)
$H_1$: the means between group are not equal(there's difference)
$\bar{X_1} - \bar{X_2} \sim N(\mu_1-\mu_2,\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2})$ if we know the population variance
$t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}~\sim t_{d''}$
$d' = round(d'') = \frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})^2}{\frac{s_1^2}{n_1}^2/(n_1-1)+\frac{s_2^2}{n_2}^2/(n_2-1)}$
$Reject ~ H_0 ~ if ~ |t|>t_{df,1-\alpha/2}$
$Fail ~ reject ~ H_0 ~ if ~ |t|<t_{df,1-\alpha/2}$
$CI = (\bar{X_1}-\bar{X_2} ~ - ~ t_{df,1-\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}},\bar{X_1}-\bar{X_2} ~ + ~ t_{df,1-\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}})$\(H_0\) : there’s no difference between groups
\(H_1\) : at least one group is different from the other groups
\(Between~Sum~of~Square = \sum_{i=1}^k\sum_{j=1}^{n_i}(\bar{y_i} - \bar{\bar{y}})^2=\sum_i^kn_i\bar{y_i}^2-\frac{y_{..}^2}{n}\)
\(Within~Sum~of~Square = \sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y_i})^2=\sum_i^k(n_i-1)s_i^2\)
\(Between~Mean~Square = \frac{\sum_{i=1}^k\sum_{j=1}^{n_i}(\bar{y_i} - \bar{\bar{y}})^2}{k-1}\)
\(Within~Mean~Square = \frac{\sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y_i})^2}{n-k}\)
\(F_{statistics} = \frac{Between~Mean~Square}{Within~Mean~Square} \sim F(k-1,n-k)\)
\(Reject ~ H_0 ~ if ~ F>F_{k-1,n-k,1-\alpha}\)
\(Fail ~ reject ~ H_0 ~ if ~F<F_{k-1,n-k,1-\alpha}\)
$H_0$ : there's no difference between groups
$H_1$ : at least one group is different from the other groups
$Between~Sum~of~Square = \sum_{i=1}^k\sum_{j=1}^{n_i}(\bar{y_i} - \bar{\bar{y}})^2=\sum_i^kn_i\bar{y_i}^2-\frac{y_{..}^2}{n}$
$Within~Sum~of~Square = \sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y_i})^2=\sum_i^k(n_i-1)s_i^2$
$Between~Mean~Square = \frac{\sum_{i=1}^k\sum_{j=1}^{n_i}(\bar{y_i} - \bar{\bar{y}})^2}{k-1}$
$Within~Mean~Square = \frac{\sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y_i})^2}{n-k}$
$F_{statistics} = \frac{Between~Mean~Square}{Within~Mean~Square} \sim F(k-1,n-k)$
$Reject ~ H_0 ~ if ~ F>F_{k-1,n-k,1-\alpha}$
$Fail ~ reject ~ H_0 ~ if ~F<F_{k-1,n-k,1-\alpha}$\(H_0 :p_{1j} =p_{2j}=...=p_{ij}\) the proportion among \(group_i\) are equal …
\(H_1\) : For at least one column there’re two row i and i’ where the proability are not the same.
\(\mathcal{X}^2 = \sum_i^{row}\sum_j^{col}\frac{(n_{ij}-E_{ij})^2}{E_{ij}} \sim \mathcal{X}^2_{df = (row-1)\times(col-1)}\)
\(Reject ~ H_0 ~ if ~ \mathcal{X}^2>\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}\)
\(Fail ~ reject ~ H_0 ~ if ~\mathcal{X}^2<\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}\)
$H_0$ :p_{1j} =p_{2j}=...=p_{ij}$ the proportion among $group_i$ are equal ...
$H_1$ : For at least one column there're two row i and i' where the proability are not the same.
$\mathcal{X}^2 = \sum_i^{row}\sum_j^{col}\frac{(n_{ij}-E_{ij})^2}{E_{ij}} \sim \mathcal{X}^2_{df = (row-1)\times(col-1)}$
$Reject ~ H_0 ~ if ~ \mathcal{X}^2>\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}$
$Fail ~ reject ~ H_0 ~ if ~\mathcal{X}^2<\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}$\(H_0\) : Group A and Group B are independet
\(H_1\) : Group A and Group B are dependet/associate
\(\mathcal{X}^2 = \sum_i^{row}\sum_j^{col}\frac{(n_{ij}-E_{ij})^2}{E_{ij}} \sim \mathcal{X}^2_{df = (row-1)\times(col-1)}\)
\(Reject ~ H_0 ~ if ~ \mathcal{X}^2>\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}\)
\(Fail ~ reject ~ H_0 ~ if ~\mathcal{X}^2<\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}\)
$H_0$ : Group A and Group B are independet
$H_1$ : Group A and Group B are dependet/associate
$\mathcal{X}^2 = \sum_i^{row}\sum_j^{col}\frac{(n_{ij}-E_{ij})^2}{E_{ij}} \sim \mathcal{X}^2_{df = (row-1)\times(col-1)}$
$Reject ~ H_0 ~ if ~ \mathcal{X}^2>\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}$
$Fail ~ reject ~ H_0 ~ if ~\mathcal{X}^2<\mathcal{X}^2_{(r-1))*(c-1),1-\alpha}$