Question

items is inspected ppose that every item in a lot of N 3 small manufactured for defects in workmanship and manufacture. The inspection results are as follows: Item: BC No. of defects: 0 1 2 (a) The number of defects is the variable Y of interest. Find the population mean u (the average numbor of defiects per item) and the population total (totai 0o) We selet a sample te number ofdefects). of the estimators for sarnple mean y, population total , and variance s2. (c) Calculate E(j), E(f), and E(82). (d) Estimate the variance of T.

Answer 1

$(a)~\mu=(0+1+2)/3=1,~population~total=0+1+2=3\\ (b)~\\ \begin{matrix} Possible~Sample & Sample~values& Value~of~sample~mean&total ~(\hat\tau)&s^2=\frac{(y_1-y_2)^2}{2}\\ (A,A)& (0,0) &0&0&0 \\ (A,B) & (0,1) &0.5&1 &0.5\\ (A,C) &(0,2) &1&2&2 \\ (B,A) &(1,0) &0.5&1&0.5 \\ (B,B)& (1,1) & 1&2&0\\ (B,C)& (1,2) &1.5&3&0.5 \\ (C,A) &(2,0) &1 &2&2\\ (C,B)& (2,1)&1.5&3 &0.5\\ (C,C)& (2,2) &2&4&0 \\ \end{matrix}\\ (c)~Sampling~distribution~of~\bar{y}~in~SRSWR\\ \begin{matrix} Value~of~\bar{y} &value~of~\hat{\tau}&Probability \\ 0&0&1/9 \\ 0.5 &1&2/9 \\ 1&2&3/9 \\ 1.5&3& 2/9\\ 2&4&1/9 \end{matrix}\\ \begin{matrix} Value~of~s^2 & Probability\\ 0 &3/9 \\ 0.5 & 4/9\\ 2& 2/9 \end{matrix}\\$

$E(\bar{y})=0*(1/9)+0.5*(2/9)+1*(3/9)+1.5*(2/9)+2*(1/9)=1\\ E(\hat{\tau})=0*(1/9)+1*(2/9)+2*(3/9)+3*(2/9)+4*(1/9)=2\\ E(s^2)=0*(3/9)+0.5*(4/9)+2*(2/9)=2/3\\ (d)~Estimate~the~variance~of~\hat{\tau}=2^2s^2=4s^2$