Armitage and Berry $(1994,$ p$.111){?}^2$ report data on the weight gain of $19$ feraiale rats

hetween $28$ and $84$ days after thirth. The rals were pliuxyl in randomizox manner on dietsi

with high $(12$ animals$)$ and low $(7$ animals$)$ probein content.

We want to tent the hypothesin on dietary ellest: Disl a luw protesin dist realt in a

signilicantly luwes wright gain?

luvel, bat we will not go there Instesal, we'll do the ust the Baysian way uning a Githtw

manpler.

Assume that the high protein diet mosasurements $y_6,i=1,$dots,$12$ are suming from a

nornal distribution $N({}_1,\frac{1}{n}),$ where: $n$ is the prexision paraneter,

$f(n_i|{}_1,n){}_1^{\frac{1}{2}}$exp$\{\frac{}{2}(y_1-{}_1{)}^2\},i=1,$dots,$12.$

'The: low protein diet masuremens $y_k,i=1,$dots,$7$ are cxung from normal distritution

$N({}_2,\frac{1}{{}_2}),$

$f(y_a|{}_2,{}_2){}_2^{\frac{1}{2}}$exp$\{-\frac{{}_2}{2}(y_{2i}-{}_2{)}^2\},i=1,$dots,$7.$

Ascume: that ${}_1$ ind ${}_2$ have normal priors $N({}_{10},\frac{1}{{}_{10}})$ and $N({}_{an},\frac{1}{{}_{2n}}),$ resquex:tively. Thake:

prior mesus as ${}_{10}={}_{30}=110($a priori no preferemone$)$ and prexisions as ${}_{10}={}_{20}=\frac{1}{100}.$

Assume that ${}_1$ and ${}_2$ hawe gamun $Ga(a_1,b_1)$ and $Ga(a_2,b_2)$ priors with shapes $a_1=$

$a_2=0\mathrm{.}01$ and rahes $b_1=b_2=4.$

$($a$)$ Constract Gibhs sampler that will sample: ${}_1,{}_1,{}_2$ and ${}_2$ from thesir positeriors.

$($b$)$ Find sample dillerences ${}_1-{}_2.$ I'roportion of powitiwe dillerencesi approximales the:

pesiterior probahility of hypothesis $H_0={}_1>{}_2.$ What is this proportion if the number of

simulativos is $10,000,$ after burn in of $500?$

conLain $0?$