Question

Suppose the lengths in inches of a particular species of animal are normally distributed with an unknown mean μ but a known variance of 36 inches². The prior distribution of μ is assumed to be normal with mean θ = 32 inches and variance = 25 inches². Ten species are analyzed, and their lengths are 19, 26, 18, 26, 31, 31, 32, 36, 33, and 21 inches.

a. Find the posterior distribution of the mean animal (species) length, p(μ | data), and provide the values of the posterior mean, μ^*, and posterior variance, σ^2*.

b. Find the posterior probability that the average animal (species) length is greater than 35 inches.

c. Is this an observational study or randomize experiment?

Answer 1

1.) Posterior distribution of the mean animal (species) length is 19, 26, 18, 26, 31, 31, 32, 36, 33, and 21 = 273/10 = 27.3

The Poseterior variance for this sample is calculated by subtracting every value from the mean and by taking a cumulative mean square average of the errors and then dividing it by n-1 so the variance is 39.5667

2.) Posterior probability that the average animal (species) length is 27.3/32 is 0.85

3.) This is an observational study because the data was collected and summarised for further analysis