# Question 1:

## Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $X \sim B(n, p)$ | | |
| Let $p$ = probability of a 'no-show' (for population) | B1 | Definition of $p$ must include word probability (or chance or proportion or percentage or likelihood but NOT possibility). Allow $p$ = P(no-show) for B1 |
| $H_0: p = 0.15$ | B1 | Allow $p=15\%$, allow $\theta$ or $\pi$ and $\rho$ but not $x$. However allow any single symbol if defined. Allow $H_0 = p=0.15$. Do not allow $H_0: P(X=x) = 0.15$. Do not allow $H_0$: $=0.15$, $=15\%$, $P(0.15)$, $p(0.15)$, $p(x)=0.15$, $x=0.15$ (unless $x$ correctly defined as a probability). Do not allow $H_0$ and $H_1$ reversed for B marks but can still get E1 |
| $H_1: p < 0.15$ | B1 | Do not allow $H_1: p \leq 0.15$. Allow NH and AH in place of $H_0$ and $H_1$ |
| $H_1$ has this form because the hospital management hopes to reduce the proportion of no-shows | E1 | Allow correct answer even if $H_1$ wrong. For hypotheses given in words allow Maximum B0B1B1E1. Hypotheses in words must include probability (or chance or proportion or percentage) and the figure 0.15 oe. |

## Part (iii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \leq 1) = 0.1756 > 5\%$ | M1 | For probability seen, but not in calculation for point probability |
| | M1 dep | For comparison with 5% |
| So not enough evidence to reject $H_0$. Not significant. | A1 | Allow accept $H_0$, or reject $H_1$. Zero for use of point prob $P(X=1) = 0.1368$. Do NOT FT wrong $H_1$ |
| Conclude that there is not enough evidence to indicate that the proportion of no-shows has decreased | E1 dep | Full marks only available if 'not enough evidence to...' mentioned somewhere. Do not allow 'enough evidence to reject $H_1$' for final mark but can still get 3/4. Upper end comparison: $1 - 0.1756 = 0.8244 < 95\%$ gets M2 then A1E1 |

**Note: use of critical region method scores**
- M1 for region $\{0\}$
- M1 for 1 does not lie in critical region, then A1 E1 as per scheme

**Line diagram method:**
- M1 for squiggly line between 0 and 1 with arrow pointing to left
- M1 for 0.0388 seen on diagram from squiggly line or from 0
- A1E1 for correct conclusion

**Bar chart method:**
- M1 for line clearly on boundary between 0 and 1 with arrow pointing to left
- M1 for 0.0388 seen on diagram from boundary line or from 0
- A1E1 for correct conclusion

## Part (iv)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $6 < 8$ | M1 | For comparison seen. Allow '6 lies in the CR'. Do NOT insist on 'not enough evidence' here |
| So there is sufficient evidence to reject $H_0$ | A1 | Do not FT wrong $H_1: p > 0.15$ but may get M1 |
| Conclude that there is enough evidence to indicate that the proportion of no-shows appears to have decreased | E1 | For conclusion in context. In part (iv) ignore any interchanged $H_0$ and $H_1$ seen in part (ii) |

**Total for part (iv): 3 marks**

## Part (v)

| Answer | Marks | Guidance |
|--------|-------|----------|
| For $n \leq 18$, $P(X \leq 0) > 0.05$ so the critical region is empty | E1 | E1 also for sight of 0.0536. Condone $P(X=0) > 0.05$ or all probabilities or values (but not outcomes) in table (for $n \leq 18$) $> 0.05$. Or 'There is no critical region'. For second E1 accept '$H_0$ would always be accepted' |
| | E1 indep | For critical region is empty. Do NOT FT wrong $H_1$. Use professional judgement - allow other convincing answers |

**Total for part (v): 2 marks**

**TOTAL: 18 marks**

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