The Brambleford vaccine batch

Step 1The batch

Brambleford, a vaccine maker, must check a new batch before it is released. From earlier batches, a dose is twice as likely to work satisfactorily as not, so the probability p that a dose is unsatisfactory solves \(p + 2p = 1\): \(p = \tfrac{1}{3}\). The quality team looks at the batch three ways: 10 doses, then 20 doses for a city's health workers, then the time 50 doses take to work.

Step 2Ten doses

Each dose works or not, independently, with the same p, and the team counts the unsatisfactory ones among 10: \(X \sim B(10, \tfrac{1}{3})\). Keep p as the fraction 1/3; rounding it to 0.33 shifts every answer (chapter 5).

Step 3Twenty doses for a city

A city asks for 20 doses for its health workers. Now \(X \sim B(20, \tfrac{1}{3})\), with a mean of \(\tfrac{20}{3} = 6.67\) unsatisfactory doses.

Step 4The time a dose takes to work

Fifty volunteers each receive a dose (the lab's dose-times.csv). Their times have a mean of 13.39 hours and a standard deviation of 4.76 hours, and the histogram is roughly a bell, so the team models a dose's time as \(N(13.39, 4.76)\).

Step 5The mean time, with its uncertainty

The model describes single doses. For the batch's mean time to effect, chapter 11's t interval applies: with n = 50, \(t^* = 2.0096\), and the 95 % interval is 13.39 ± 1.35 hours, from 12.04 to 14.74 hours. Single doses vary far more: the model puts one dose in ten beyond 19.49 hours.