Minitab Hints for problem 7.22

In problem 7.22 you are asked to compute a "95% posterior probability interval". By that we mean a range of values that has these two features: 1) the probability that the unknown proportion is in the range is 95%; and 2) the probability that the unknown proportion is outside the range on the right is the same as the probability that the unknown proportion is outside the range on the left.

To find this range, or interval, you need to compute percentiles. What is a percentile? The 10% (or .1) percentile of a random variable is the point that has 10% of the probability to its left (and the rest to its right). One percentile you already know about is the median, which is the 50% percentile. For another example, look at Figure 7.18 in Berry, bottom panel. The probability to the left of .5 is 6%, so we say that .5 is the 6% (or .06) percentile.

To get a 95% posterior probability interval you need to compute two percentiles: the .025 percentile and the .975 percentile. This is because you want 95% of the probability to be inside the range and the rest to be outside, and also evenly divided between left and right. If this sounds puzzling, sketch the picture of a distribution and try to identify the two percentiles we are talking about.

To do this calculation precisely with minitab use the macro "p_beta" and follow the steps on pages 234 and 235 in Berry, until you get to "type y to compute percentiles". Type "y" and then type 

.025 .975
at the "data" prompt immediately following. Minitab will give you two numbers that are the two percentiles and are also the endpoints of the 95 percent posterior probability interval you are looking for. Remember that the values 5.5 and 5 used in the example in the book are not necessarily the ones you need for problem 7.22.

Section 7.5 also describes an alternative approach to doing this calculation, based on an approximation that was popular before we had computers. Don't bother with that unless you are curious about it, or minitab refuses to work for you.