91Ó°ÊÓ

According to Example 7.3.10, 20 nationally prepared food were sampled and their stated calorie contents were compared with the calorie contents determined in the laboratory. The conditional distribution of these differences is assumed to have a normal distribution with a mean \(\theta \)and a variance\({\sigma ^2} = 100\). The sample mean is\({\bar x_n} = 0.125\). The prior distribution of \(\theta \) follows a normal distribution with a mean 0, and a variance \({v^2} > 0\). The posterior mean of \(\theta \) is 0.12.

When the conditional distribution of a sample has a normal distribution with mean \(\theta \) and variance of \({\sigma ^2} = 100\) and the prior distribution of \(\theta \) follows a normal distribution with mean 0 and variance of \({v^2} > 0\), the posterior mean of \(\theta \) is defined as,

\(\begin{aligned}{\mu _1} = \frac{{{\sigma ^2}{\mu _0} + n{v_0}^2{{\bar x}_n}}}{{{\sigma ^2} + n{v_0}^2}}\\ \Rightarrow 0.12 = \frac{{100 \times 0 + 20 \times {v^2} \times 0.125}}{{100 + 20{v^2}}}\\ \Rightarrow 12 + 2.4{v^2} = 2.5{v^2}\\ \Rightarrow {v^2} = 120\end{aligned}\)

Hence, \({v^2} = 120\) was used.