OCR MEI S1 — Question 3 19 marks

Exam BoardOCR MEI
ModuleS1 (Statistics 1)
Marks19
PaperDownload PDF ↗
TopicHypothesis test of binomial distributions
TypeMultiple binomial probability calculations
DifficultyStandard +0.3 This is a straightforward S1 binomial question requiring standard calculations (direct probability, cumulative probability, and a basic hypothesis test). All parts use routine methods with no novel problem-solving required, though part (iv) requires systematic trial of values. Slightly above average due to the multi-part nature and the hypothesis test component, but all techniques are standard textbook exercises.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail
3 At a doctor's surgery, records show that \(20 \%\) of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients. There are 16 appointments to see the doctor one afternoon.
  1. Find the probability that all 16 patients turn up.
  2. Find the probability that more than 3 patients do not turn up. To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.
  3. The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.
  4. Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up. A computerised appointment system is introduced at the surgery. It is decided to test, at the \(5 \%\) level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.
  5. Write down suitable hypotheses and carry out the test.
3 At a doctor's surgery, records show that $20 \%$ of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients.

There are 16 appointments to see the doctor one afternoon.\\
(i) Find the probability that all 16 patients turn up.\\
(ii) Find the probability that more than 3 patients do not turn up.

To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.\\
(iii) The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.\\
(iv) Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up.

A computerised appointment system is introduced at the surgery. It is decided to test, at the $5 \%$ level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.\\
(v) Write down suitable hypotheses and carry out the test.

\hfill \mbox{\textit{OCR MEI S1  Q3 [19]}}
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Q1 8 Q2 18 Q3 19