# Question 3:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(50000 < X < 55000) = P\!\left(\dfrac{50000 - 50600}{3400} < Z < \dfrac{55000 - 50600}{3400}\right)$ | M1 | For standardising both. SOI. Penalise erroneous continuity corrections and wrong sd. Condone numerator(s) reversed. |
| $= P(-0.176 < Z < 1.294) = \Phi(1.294) - (1 - \Phi(0.176)) = 0.9022 - 1 + 0.5699$ | M1 | For correct structure $\Phi(\text{positive } z) - \Phi(\text{negative } z)$ |
| $= 0.4721$ | A1 | CAO including use of difference tables (Answer from calculator 0.4722 and from tables interpolated 0.4723) |
| **[3]** | | |

## Question 3(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X > 45000) = P\left(Z \geq \frac{750-751.4}{2.5}\right)\left(Z > \frac{45000-50600}{3400}\right) = P(Z > -1.647)$ | B1* | For $-1.647$ or $-\Phi^{-1}(0.95) = -1.645$ or $1.647$ seen with $P(X < 56200)$ or numerator reversed |
| $= \Phi(1.647) = 0.9502$ | B1* | For $0.9502$ or $45007$ or $0.0498$, or B1 for $-1.645$ if B1 for $-1.647$ already awarded |
| $0.9502 > 95\%$ so agree with claim | depE1* | **For comparison seen** e.g. $-1.647 < -1.645$ or $0.0498 < 0.05$ or $1.647 > 1.645$ or $95\%$ last longer than 45007 hours, **and correct conclusion**. Dependent on B1, B1 awarded |

## Question 3(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| From tables $\Phi^{-1}(0.999) = 3.09$ | B1 | $\pm 3.09$ seen |
| $\frac{h - 50600}{3400} = -3.09$ | M1 | For equation with their negative z-value |
| $k = 50600 - (3.09 \times 3400) = 40100$ | A1 [3] | CAO Allow 40094, 40090 |

## Question 3(iv):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(Y < 60000) = 0.6 \Rightarrow P\left(Z < \frac{60000-\mu}{\sigma}\right) = 0.6$ | B1 | For $\pm 0.2533$ or $\pm 1.282$ seen |
| $\Rightarrow \frac{60000-\mu}{\sigma} = \Phi^{-1}(0.6) = 0.2533$ | M1 | For an equation in $\mu$, $\sigma$, $z$ and $y$ formed. NB using $z = \pm 0.2533$ with $y = 60000$ or $\pm 1.282$ with $y = 50000$ |
| $\Rightarrow 60000 = \mu + 0.2533\sigma$ | | |
| $P(Y > 50000) = 0.9 \Rightarrow P\left(Z > \frac{50000-\mu}{\sigma}\right) = 0.9$ | | |
| $\Rightarrow \frac{50000-\mu}{\sigma} = \Phi^{-1}(0.1) = -1.282$ | A1 | For two correct equations seen |
| $\Rightarrow 50000 = \mu - 1.282\sigma$ | | |
| $1.5353\sigma = 10000$, $\sigma = 6513$ | A1 | CAO Allow 6510, 6515 |
| $\Rightarrow \mu = 50000 + (1.282 \times 6513) = 58350$ | A1 [5] | CAO Allow 58400 |

## Question 3(v):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Two Normal curve shapes shown | G1 | For two Normal shapes including attempt at asymptotic behaviour with horizontal axis at each of the four ends. Penalise clear asymmetry |
| Means shown explicitly or by scale **on a single diagram** | G1 | If shown explicitly, positions must be consistent with horizontal scale if present. FT part (iv) |
| Greater width (variance) for Different model | G1 | FT part (iv) |
| Lower max height for Different model | G1 [4] | FT part (iv). If not labelled assume the larger mean represents Different model |