In a school where all the teachers drink coffee in the staffroom, a headteacher wants to see if using a new brand of coffee in the staffroom improves teachers’ report writing punctuality. Previously, 75% of teachers in her school would meet the deadline for writing reports. After the new brand of coffee is introduced the headteacher takes a random sample of 15 teachers, and conducts an experiment to test whether the proportion of teachers meeting the deadline on the next set of reports has improved. The test is conducted at the 5% significance level, using the following hypotheses:
(i)
Write down a suitable distribution for the random variable
, the number of teachers in the headteacher’s sample who meet the deadline.
(ii)
Calculate the probability that all the teachers in the sample would meet the deadline if the null hypothesis were true.
(iii)
By first calculating the probability that exactly 14 of the teachers would meet the deadline if the null hypothesis were true, find the critical value for the test.