Question

A team of biologists studying the effects of warming ocean on fish health captured and released a sample of fish and measured the length of each captured fish. Researchers modeled lengths of a given species of fish as independent and identically distributed normal random variables with unknown mean Q and known standard deviation. The sample contained twenty members of the species. The lengths, measured in millimeters, were: 56, 59, 55, 55, 57, 61, 58, 56, 61, 53, 51, 59, 62, 54, 54, 58, 59, 56, 62, 56. Based on prior knowledge of similar fish species, researchers chose a normal prior distribution for Q with prior mean of 45 centimeters and standard deviation 7 centimeters. For purposes of this analysis, assume the known standard deviation is equal to the sample standard deviation of the measured fish lengths. (This simplifying assumption is not unreasonable for a sample this size if we are interested only in inferences about the mean.) Find the posterior distribution of the mean fish length Q given the observations. Name the distribution type and hyperparameters. What is the predictive distribution for the measured length of the next fish captured of the same species? State the distribution type and the hyperparameters. Find a 95% credible interval on the measured length of the next fish.

Answer 1

To find the posterior distribution of the mean fish length Q given the observations, we can use Bayes' theorem and combine a normal prior distribution with a normal likelihood distribution. The predictive distribution for the measured length of the next fish captured can be obtained by taking the average of the posterior distribution. To find a 95% credible interval on the measured length of the next fish, we can use the quantiles of the posterior distribution.

Step-by-step explanation:

To find the posterior distribution of the mean fish length Q given the observations, we can use Bayes' theorem. The prior distribution of Q is given as a normal distribution with mean 45 centimeters and standard deviation 7 centimeters. The likelihood of the observed data can be modeled as a normal distribution with mean Q and standard deviation equal to the sample standard deviation of the measured fish lengths. By combining the prior distribution and the likelihood, we can calculate the posterior distribution of Q.

The predictive distribution for the measured length of the next fish captured can be obtained by taking the average of the posterior distribution. The distribution type of the posterior distribution is also a normal distribution, with updated hyperparameters based on the observed data. However, to provide an accurate prediction, we also need to take into account the uncertainty in our estimation. This can be done by calculating a credible interval, which represents a range of values within which we can be confident the true fish length falls at a certain confidence level.

To find a 95% credible interval on the measured length of the next fish, we can use the quantiles of the posterior distribution. The interval will stretch from the lower quantile to the upper quantile that encompass 95% of the probability mass. This interval gives us a range of plausible values for the fish length, considering both the prior information and the observed data.