There is some debate regarding advertised nutrition and caloric content of restaurant foods. L. E. Urban and colleagues investigated the advertised caloric content of food items at 29 fast-food and sit-down restaurants and found that the measured caloric content of the food items in a laboratory setting tended to be higher than the reported or advertised values.12 Suppose that the differences between the laboratory measured calories and the advertised calorie values are taken to be normal with unknown mean difference m calories and variance σ2 = 1000 calories2, and assign a normal prior distribution to µ with mean 0 (no difference, on average, between the measured and advertised caloric content) and variance σ2 = 10,000. The actual sample mean of the differences between the measured and advertised calorie values was 25.8 calories. Based on the data reported in the article, we’ll use a Bayesian analysis to examine whether restaurants are systematically underreporting the caloric content of their food items.
a. Describe the posterior distribution of µ, p(µ | data), and provide the posterior mean and variance.
b. Use your posterior distribution to find the probability that, on average, the measured caloric con-tent is at least 10 calories more than the advertised caloric content of restaurant food items; that is, find p(µ > 10 | data). Does your answer support the argument that restaurants are systematically underreporting caloric content?
c. A 95% Bayesian credible interval can also be constructed for m by finding the limits L and U such that P(µ < L | data) = 0.025 and P(µ > U | data) = 0.025. Construct a 95% credible interval for the true mean difference in caloric content, and provide an interpretation of the interval.