The following data come from the experiment reported by MacGregor et al. (1979). They give the supine systolic blood pressures (mm Hg) for fifteen patients with moderate essential hypertension. The measurements were taken immediately before and two hours after taking a drug.  

<html><body><table><tr><td>Patient</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td>Before After</td><td>210 201</td><td>169 165</td><td>187 166</td><td>160 157</td><td>167 147</td><td>176 145</td><td>185 168</td><td>206 180</td></tr><tr><td>Patient</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td></td></tr><tr><td>Before After</td><td>173 147</td><td>146 136</td><td>174 151</td><td>201 168</td><td>198 179</td><td>148 129</td><td>154 131</td><td></td></tr></table></body></html>  

We are interested in the effect of the drug on blood pressure. We assume that, given parameters $\mu,~\tau$ , the changes in blood pressure, from before to after, in the $n$ patients are independent and normally distributed with unknown mean $\mu$ and unknown precision $\tau$ . The fifteen differences are as follows.  

Our prior distribution for $\tau$ is a gamma(0.35, 1.01) distribution. Our conditional prior distribution for $\mu$ given $\tau$ is a normal $N(0,\ [0.003\tau]^{-1})$ distribution.  

(a) Find the marginal posterior distribution of $\tau$ .  
(b) Find the marginal posterior distribution of $\mu$ .  
(c) Find the marginal posterior 95% hpd interval for $\mu$ .  
(d) Comment on what you can conclude about the effect of the drug. 