What is the main reason the council cannot conclude from Output 1.1 that high pollution causes asthma surges?

From Output 1.1, P̂(High | Surge) = 36/60 = 0.60. In the council's setting, what does this value measure?

Use the outputs provided to explain how the sensor's measurement error behaves across pollution levels. What does this imply for using the fixed cut-off of 25 µg/m³ on the sensor to flag High days, and what would you recommend?

Using Output 1.1, what is the estimated probability that a day is a surge day, given that the sensors flagged it as High?

Suppose we decide to run an MCMC to sample from the posterior of a binomial proportion theta. They use the MCMCpack package and run the MCMCmetrop1R() function, with a burnin = 1000, starting the chain at 0.5. 

What does the burnin = 1000 mean?

Suppose the university now believes enrolment is likely to be high, and expresses this as a Normal(0.8, 0.1) prior on p, truncated to the interval (0, 1). This prior is not conjugate to the binomial likelihood, so the posterior cannot be found with a Beta update and is approximated by simulation using MCMCpack::MCMCmetrop1R(). This function requires the log of the unnormalised posterior. Of the 45 students surveyed, 31 said "yes".

Which of the following should replace the blank line?

log_post <- function(theta) {
  if (theta <= 0 || theta >= 1) return(-Inf)
  ____________________
}

Using MCMCmetrop1R():

• draw 5,000 samples from the posterior distribution of p, 
• starting the chain at 0.5, 
• with a burn-in of 1,000 and 
• the seed 24205242. 

Use these samples determine a 95% credible interval for p.

A student runs a Metropolis sampler to estimate the posterior of a binomial proportion, starting the chain at 0.5. The posterior is centred near 0.72, so the starting value sits well out in the lower tail. The student decides not to use any burn-in and keeps every iteration, including the very first ones.

What happens as a result?

Quantile    1%      2.5%    5%      25%     50%     75%     95%     97.5%   99%
Value       0.5953  0.6141  0.6330  0.6882  0.7249  0.7606  0.8097  0.8240  0.8391

The university additionally surveyed 45 students who were eligible to take the proposed Sports Analytics elective. Each student was asked whether they would enrol if the course were offered (either "yes" or "no").

Which distribution is most appropriate for modelling the number of students, out of the 45 surveyed, who say they would enrol?