두 모비율 간 차이 구간추정

\begin{align*} &A \kern{10pt} \longrightarrow \kern{10pt} X_i \sim bernoulli(p_1) \\ &B \kern{10pt} \longrightarrow \kern{10pt} X_i \sim bernoulli(p_2) \\[10pt] &\therefore~-1 < p_1 -p_2 < 1 \end{align*}

\begin{align*} f_{\underline{X},\underline{Y}} (\underline{x},\underline{y}) &= f_{\underline{X}}(\underline{x}) f_{\underline{Y}}(\underline{y}) = \prod_{i=1}^{n} f_{X_{i}}(x) \prod_{i=1}^{n} f_{Y_{i}}(y) \\[15pt] f_{\underline{X}}(\underline{x}) &= {p_1}^{x_i}~(1-p_i)^ {1-x_1}~I(x_i \in \{0,1\}) \\[10pt] f_{\underline{Y}}(\underline{y}) &= {p_2}^{y_j}~(1-p_i)^ {1-y_j}~I(y_j \in \{0,1\}) \\[10pt] f_{\underline{X},\underline{Y}} (\underline{x},\underline{y}) &= {p_1}^{\sum x_i}~ (1-p_i)^{n1-\sum x_1}~ {p_2}^{\sum y_j}~ (1-p_i)^{n2-\sum y_j}~ \\[15pt] l(p_1,p_2) &= \textstyle \sum x_i \log p_1 ~+~ (n1-\sum x_i)\log (1-p_1) ~+~ \\[5pt] &~ + \textstyle \sum y_j \log p_2 ~+~ (n2-\sum y_j) \log (1-p_2) \\[10pt] &= \textstyle n1 \overline{X}\log p_1 ~+~ n1(1-\overline{X})\log (1-p_1) ~+~ \\[5pt] &~ + \textstyle n2 \overline{Y}\log p_2 ~+~ n2(1-\overline{Y})\log (1-p_2) ~+~ \\[15pt] {\partial \over \partial p_1}~ l(p_1,p_2) &= \dfrac{n_1 \overline{X}}{p_1} - \dfrac{n_1 (1-\overline{X})} {1-p_1} \\[15pt] &= \dfrac{n_1}{p_1(1-p_1)}~ \{(1-p_1)\overline{X} - (1-\overline{X})p_1\} \\[15pt] &= \dfrac{n_1}{p_1(1-p_1)} (\overline{X} - p_1) = 0 \end{align*}

\begin{align*} \therefore \kern{8pt} &\hat p_1 = \overline{X}, \kern{10pt} \hat p_2 = \overline{Y} \\ &1- \alpha = P(a < \hat p_1 - \hat p_2 < b) \end{align*}

\begin{align*} &X_i \sim bernoulli(p_1) \\[5pt] &E(X_i) = p_1. \kern{10pt} Var(X_i) = p_1(1-p_1) \\[10pt] &\dfrac{\sqrt{n1}~ (\overline{X}-p_1)} {\sqrt{p_1(1-p_1)}} \kern{10pt} \longrightarrow \kern{10pt} N(0,1) \end{align*}

\begin{align*} &Y_i \sim bernoulli(p_2) \\[5pt] &E(Y_i) = p_1. \kern{10pt} Var(Y_i) = p_1(1-p_1) \\[10pt] &\dfrac{\sqrt{n2}~ (\overline{Y}-p_2)} {\sqrt{p_2(1-p_2)}} \kern{10pt} \longrightarrow \kern{10pt} N(0,1) \end{align*}

\hat p_1 - \hat p_2 = \overline{X} - \overline{Y}. \kern{15pt} \text{가정 추가 : }n_1=n_2=n \\[10pt] \begin{align*} W_i &= X_i - Y_i ~~~~\text{are independent} \\ E(W_i) &= p_1-p_2 \\ Var(W_i) &= Var(X_i) + Var(Y_i) \\ &= P_1(1-P_1) + P_2(1-P_2) \\[10pt] &\dfrac{\sqrt{n}~ \big\{\overline{X}- (p_1-p_2)\big\}} {\sqrt{p_1(1-p_1)+p_2(1-p_2)}} \kern{10pt} \longrightarrow \kern{10pt} N(0,1) \end{align*}

\begin{align*} & P(a < \hat p_1 - \hat p_2 < b) =P(a<w<b) \\[10pt] &\backsimeq P \bigg( -Z_{\alpha/2} < \dfrac{\sqrt{n}~ \big\{\overline{X}- (p_1-p_2)\big\}} {\sqrt{p_1(1-p_1)+p_2(1-p_2)}} < Z_{\alpha/2} \bigg) \end{align*}

\overline{W} \pm Z_{\alpha/2}~ \sqrt{{p_1(1-p_1)+p_2(1-p_2)} \over n} \\[10pt] \overline{X}-\overline{Y} \pm Z_{\alpha/2}~ \sqrt{{p_1(1-p_1)+p_2(1-p_2)} \over n}

\overline{W} \pm Z_{\alpha/2}~ \sqrt{{p_1(1-p_1)+p_2(1-p_2)} \over n} \ge \overline{W} \pm Z_{\alpha/2}~ {\sqrt{1}\over 2n}