## Description

1. From the class survey, y = 12 out of n = 70 sampled students had pets. R Example 8.1

(ex8.1.R, posted under Lecture Materials) illustrates how to approximate the posterior

mean of the population proportion π of people like us who have pets. It assumes a binomial

model and Jeffreys prior. Using the same binomial model and Jeffreys prior, you will

approximate the posterior variance of π.

(a) [2 pts] Write out all of the mathematical formulas for the integrals you will compute

using R.

(b) [2 pts] Perform the integrations using function integrate() in R. (You may either use

the exact value for the posterior mean of π, or approximate it using integrate().)

(c) [2 pts] Now compute the posterior variance analytically (with the help of conjugacy

and Table A.2 in Cowles), and compare this answer to your approximation.

2. Official combined city/highway energy consumption ratings (MPGe, miles per gallon

gasoline equivalent) are given below for 2019 model all-electric vehicles in two categories:1

Small Cars

Hyundai Ioniq Electric 136

Volkswagen e-Golf 119

Honda Clarity EV 114

BMW i3 113

BMW i3s 113

Nissan Leaf 112

Fiat 500e 112

smart EQ fortwo (coupe) 108

smart EQ fortwo (convertible) 102

Sport Utility Vehicles (SUVs)

Hyundai Kona Electric 120

Tesla Model X 75D 93

Tesla Model X 100D 87

Tesla Model X P100D 85

Jaguar I-Pace 76

Regard MPGe as independent between vehicles and normally-distributed within category,

with both mean and variance possibly differing by category. Use “independent” “standard”

(product-Jeffreys) priors, as illustrated in R Example 8.3 (ex8.3.R, posted under Lecture

Materials).

Use at least 100000 simulation samples for all of your approximations.

(a) [1 pt] Compute the sample means and sample standard deviations for the two

categories.

(b) [2 pts] Compute an approximate 95% equal-tailed credible interval for the difference

between the mean for small cars and the mean for SUVs. Do the means appear to differ?

(c) [1 pt] Approximate the posterior probability that the mean for small cars does not

exceed the mean for SUVs.

1Data from https://fueleconomy.gov

1

(d) [3 pts] Compute the (frequentist) Welch two sample t-test one-sided p-value for testing

the null hypothesis that the mean for small cars does not exceed the mean for SUVs.

(Use R function t.test(…, …, alternative=…, var.equal=FALSE), making

sure to select the correct alternative.) Also compute the usual (frequentist) two sample

t-test one-sided p-value that assumes equal variances (t.test(…, …,

alternative=…, var.equal=TRUE)). Compare with the Bayesian probability of the

previous part.

(e) [2 pts] Compute an approximate 95% equal-tailed credible interval for the ratio of the

variance for small cars to the variance for SUVs. Do the variances appear to differ?