June 2022 Paper 3 Q4
4. A dentist knows from past records that 10% of customers arrive late for their appointment.
A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment.
A random sample of 50 of the dentist’s customers is taken.
- a null hypothesis corresponding to no change in the proportion of customers who arrive late
- an alternative hypothesis corresponding to the manager’s belief
You should state the probability of rejection in each tail, which should be less than 0.025 (3)
The manager observes that 15 of the 50 customers arrived late for their appointment.
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0: p = 0.1 \qquad \mathrm{H}_1: p \ne 0.1\) | B1 | 2.5 |
| (1) |
Notes
B1: For both hypotheses in terms of \(p\) or \(\pi\). Connected to \(\mathrm{H}_0\) and \(\mathrm{H}_1\) correctly
Condone 10% but not 10
| Scheme | Marks | AO |
|---|---|---|
| Use of \(X \sim \mathrm{B}(50,\ 0.1)\) implied by sight of one of awrt 0.0052 or awrt 0.9755 or awrt 0.0245 | M1 | 3.4 |
| Critical regions \(X = 0\) or \(X \geqslant 10\) | A1 | 1.1b |
| \(X = 0\) and \(X \geqslant 10\) plus \(\mathrm{P}(X = 0) =\) awrt 0.0052 and \(\mathrm{P}(X \geqslant 10) =\) awrt 0.0245 | A1 | 1.1b |
| SC: Both CR correct with no probabilities and no distribution seen scores M0A1A0 | ||
| (3) |
Notes
M1: Using correct distribution to find the probability associated with one tail of the CR
If the correct distribution is stated (may be seen in part(a)) allow for one tail of the correct CR or one of (awrt 0.025 or awrt 0.005 or awrt 0.975) seen connected to a correct probability statement
A1: Lower CR \(X = 0\ /\ X \lt 1\ /\ X \leqslant 0\ /\) [condone eg \(\mathrm{P}(X = 0)\) labelled as CR]
Or Upper CR \(X \geqslant 10\) or \(X \gt 9\) [condone \(\mathrm{P}(X \geqslant 10)\) oe labelled as CR]
A1: Both CR’s correct with the relevant probabilities Allow \(\cup\) for “and” and \(X \gt 9,\ X \lt 1,\ X \leqslant 0\) [do not allow \(\mathrm{P}(X = 0)\) or \(\mathrm{P}(X \geqslant 10)\) oe]
Allow CR in different form eg \((9,\ \infty)\), \([10,\ \infty)\)
| Scheme | Marks | AO |
|---|---|---|
| 0.0297 | B1ft | 1.1b |
| (1) |
Notes
B1ft: awrt 0.0297 or 2.97% or ft for the sum of the probabilities in (b) for “their 2 critical regions” if seen. If none seen it must be awrt 0.0297
SC M0 in (b) for a one tail test Allow B1ft for their one tail CR in (b) eg 0.0338 or 0.0245 or 0.0579
| Scheme | Marks | AO |
|---|---|---|
| 15 is in the critical region therefore there is evidence to support the manager’s belief | B1ft | 2.2b |
| (1) | ||
| (6 marks) |
Notes
B1ft: A correct statement about 15 and “their CR” or sight \(\mathrm{P}(X \geqslant 15) = 0.0000738\ldots\) and comparison with “their 0.0245”
and a compatible correct statement in context. eg There is evidence that there has been a change in the proportion/probability arriving late
Condone increase rather than change
Do not allow contradicting statements.
NB No CR given in (b) then B0