The Brambleford vaccine batch
Check a new batch of a vaccine: unsatisfactory doses among 10 and 20 as binomials, then the time a dose takes to work as a normal.
Draws on 5. The binomial distribution, 7. The normal distribution, 11. Student's t and confidence intervals
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).
Exactly three unsatisfactory
What is the probability that exactly 3 of the 10 doses are unsatisfactory?
Nine or ten work
What is the probability that more than 8 of the 10 doses work satisfactorily?
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.
More than half unsatisfactory
What is the probability that at least 11 of the 20 doses are unsatisfactory?
At least thirteen work
What is the probability that at least 13 of the 20 doses work satisfactorily?
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 \(N(13.39, 4.76)\).
Between 10 and 15 hours
Using the model \(N(13.39, 4.76)\), what is the probability that a dose takes between 10 and 15 hours to work?
Nine doses in ten
Using the model \(N(13.39, 4.76)\), within how many hours do 90 % of doses work (the 90th percentile, to four decimals)?
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.