91Ó°ÊÓ

For given\(AQL\)and\(LTPD\), the ratio is

\(\frac{{LTPD}}{{AQL}} = \frac{{0.07}}{{0.02}} = 3.5.\)

The closest value is to\(3.5\)is\(3.55\), so choose\(c = 5\). For such value,\(n\)is

\(n = \frac{{n{p_1}}}{{{p_1}}} = \frac{{2.613}}{{0.02}} = 130.64 \approx 131\)

Theorem:

\(b(x;n,p) = \left\{ {\begin{aligned}{*{20}{l}}{\left( {\begin{aligned}{*{20}{l}}n\\x\end{aligned}} \right){p^x}{{(1 - p)}^{n - x}}}&{,x = 0,1,2, \ldots ,n}\\0&{,{\rm{ otherwise }}}\end{aligned}} \right.\)

Cumulative Density Function cdf of binomial random variable\(X\)with parameters\(n\)and\(p\)is

\(B(x;n,p) = P(X \le x) = \sum\limits_{y = 0}^x b (y;n,p),\;\;\;x = 0,1, \ldots ,n.\)

The value of\(\alpha \)is

\(\begin{aligned}{l}\alpha = P(X > 5{\rm{ when }}p = 0.02) = 1 - \sum\limits_{x = 0}^5 {\left( {\begin{aligned}{*{20}{c}}{131}\\x\end{aligned}} \right)} {0.2^x}{(1 - 0.2)^{131 - x}}\\ = 0.487.\end{aligned}\)

The value of\(\beta \)is

\(\begin{aligned}{l}\beta = P(X \le 5{\rm{ when }}p = 0.07) = \sum\limits_{x = 0}^5 {\left( {\begin{aligned}{*{20}{c}}{131}\\x\end{aligned}} \right)} {0.7^x}{(1 - 0.7)^{131 - x}}\\ = 0.0974.\end{aligned}\)

\(\alpha = 0.487;\beta = 0.0974.\)