The psychologist Tversky and his colleagues say that about four out of five people will answer (a) to the following question: A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital 15 babies are born each day. Although the overall proportion of boys is about 50 percent, the actual proportion at either hospital may be more or less than 50 percent on any day
At the end of a year, which hospital will have the greater number of days on which more than 60 percent of the babies born were boys? (a) the large hospital (b) the small hospital *(c) neither-the number of days will be about the same. Assume that the probability that a baby is a boy is .5 (actual estimates make this more like .513). Decide, by simulation, what the right answer is to the question. Can you suggest why so many people go wrong?
#we create observations for each hospital for a single year
#each row represents a day of the year with N observations, each representing a bitrh
#we assume that an observation >= 0.5 is a boy, < 0.5 is a girl
large.hospital <- (matrix(sample(c(0:100000),size = 365*45, replace = T), nrow=365, ncol=45))/100000
small.hospital <- (matrix(sample(c(0:100000),size = 365*15, replace = T), nrow=365, ncol=15))/100000
#compute porportion of days where > 60% boys were born
large.birthrate <- sum((rowSums(large.hospital >= 0.5) / 45) > 0.6) / 365
small.birthrate <- sum((rowSums(small.hospital >= 0.5) / 15) > 0.6) / 365
#output the birthrates
large.birthrate## [1] 0.08219178## [1] 0.1561644So for a single sample (1 year), we see that the probability of > 60% boys is significantly higher at the smaller hospital.
#now let's simulate for 10K years at each hospital
large.hospital <- (matrix(sample(c(0:100000),size = 3650000*45, replace = T), nrow=365, ncol=45))/100000
small.hospital <- (matrix(sample(c(0:100000),size = 3650000*15, replace = T), nrow=365, ncol=15))/100000
#compute porportion of days where > 60% boys were born
large.birthrate <- sum((rowSums(large.hospital >= 0.5) / 45) > 0.6) / 365
small.birthrate <- sum((rowSums(small.hospital >= 0.5) / 15) > 0.6) / 365
#output the birthrates
large.birthrate## [1] 0.05753425## [1] 0.1534247If we simulate for 10K years, we see the same thing.
Answer: The small hospital is more likely to have a higher number of days with > 60% boys born. I suspect that most people get this wrong because they misinterpret the question slightly. They probably (correctly) equate larger sample size with a better estimate of the mean in question, however, here we’re not looking for a better estimate of the mean, we’re actually looking for the hospital that provides higher variability around the mean - hence fewer observations (15 births vs 45) is what we want.