## Question 10:

### Part (a):
$[=]153$ | **B1 [1]** |

### Part (b):
$z = \pm1.645$ used | **B1** | Or $\pm1.644(85\ldots)$

Their positive $1.645 = \frac{183 - their\ 153}{\sigma}$ oe $(= 18.237\ldots \text{ to } 18.248\ldots)$ | **M1** | **M0** if continuity correction used

$\sigma = 18.2$ cao | **A1 [3]** |

## Question 10(c):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{16}{452} = 0.035(398...)$ | B1 (3.1b) | Allow percentage; or B1 as main scheme then Invnorm(0.0353, their 153, their 18.2) for M1; NB 119.95 to 120.15 which is close to 120 oe for A1 |
| Their $P(X < 120)$ from $N(\text{their } 153, \text{their } 18.2^2)$ | M1 (1.1) | M0 if continuity correction used |
| Probability of 0.0349 to 0.0352 which agrees to 2 sf | A1 (2.2b) | Allow percentage |
| **Or:** B1 for their $P(X < 120)$ then M1 for $452 \times \text{their } 0.03490$, then A1 for 15.77 to 15.91 which is close to 16 oe | | |
| **Or:** B1 as main scheme then M1 for $\sigma = \frac{120-153}{\text{their}-1.809}$ and $18.3 \approx 18.2$ for A1 | | or $z = \frac{120 - \text{their }153}{\text{their }18.2}$ for M1; NB $-1.809$ to $-1.813$; Invnorm(0.0353, 0, 1) to obtain $-1.806$ to $-1.812$ which is close to $-1.809$ to $-1.813$ for A1 |

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## Question 10(d):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = \text{their } 153$; $H_1: \mu < \text{their } 153$ | B1 (1.1) | For both hypotheses; may be stated in words, but need to see 153 for B1 and population mean for 2nd B1; B0 if other parameter used unless clearly defined as population mean |
| $\mu$ is the **population mean** flight time from **Magaluf to Liverpool** | B1 (2.5) | |
| Use of $N\!\left(\text{their }153,\ \frac{\text{their }18.2^2}{24}\right)$ to find $P(\bar{X} < 143.6)$ | M1* (3.3) | Or inv Norm$(0.01, \text{their }153, \frac{\text{their }18.2^2}{24})$; or $z = \frac{143.6 - \text{their }153}{\text{their }\frac{18.2}{\sqrt{24}}}$ |
| awrt 0.0057 to 0.0058… to 2 or more sf oe **isw** | A1 (1.1) | $\bar{X} < 144$ to 144.4 is critical region; $z = -2.5248..$ to $-2.5302...$ to 2 or more dp |
| Their 0.0057 correctly compared with 0.01 oe | M1dep* (3.4) | 143.6 correctly compared with their 144.36; FT their comparison; their $z$ compared with $-2.326$ or $-2.33$ |
| Result is significant or reject $H_0$ or accept $H_1$ | A1 (1.1) | FT their comparison |
| There is sufficient evidence to **suggest** at the 1% level that the **mean** flight time from **Magaluf to Liverpool** is less than **153** / mean flight time from Liverpool to Magaluf | A1 (2.2b) | Do not allow e.g. conclude / prove / indicate or other assertive statement instead of suggest |

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## Question 10(e):

| Answer | Marks | Guidance |
|--------|-------|----------|
| **Reduce** the value of $\mu$ | B1 (3.5c) | e.g. take a bigger sample is insufficient |
| **Increase** the value of $\sigma$ or $\sigma^2$ | B1 (3.5c) | Allow B1 for e.g. use new sample data to calculate new estimate for $\sigma$ or $\sigma^2$ |

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