*normal*(30,4) and observed data with n = 5 and y = 32.

Let's change the problem a little bit: suppose the stream we sampled from is downstream of a nuclear power plant. We know that the population mean for unpolluted streams is 35 with a standard deviation of 4. We use the same variance, but since we guess that the trout are going to be smaller in this stream, we use a

*normal*(30,4) prior. And we calculate the population standard deviation for our stream from the observed values. (No value is given for sigma.x).

We want to test the one-sided hypothesis that the trout in this stream are smaller on average than normal trout. The posterior is calculated in the usual way.

We see that P(35 cm) < 0.005. We reject the hypothesis that the trout are "normal" in length. It makes some difference, but not a lot, that we used a prior mean of 30. If we had used 35, we would have

P(35 cm) < 0.01. I'll let Bolstad say it:

The posterior distribution ofg( μ | y_{1}, ..., y_{n}) summarizes ourentirebelief about the parameter, after viewing the data.

## No comments:

Post a Comment