## Question 13(a):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 0.14$, $H_1: \mu < 0.14$ | B1 | allow any other symbol except $\bar{x}$ or $\bar{X}$, as long as it is correctly defined; allow hypotheses stated in words |
| their $\mu$ is the **population mean mass** of this variety of apple | B1 | allow weight; correct definition of $\mu$ may be embedded in hypotheses written out as a sentence; do not allow $\bar{x}$ or $\bar{X}$ |

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## Question 13(b):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\bar{X} \sim\,] N\left(0.14, \frac{0.0199^2}{80}\right)$ | B1 | Normal distribution with correct mean or variance; allow variance = **awrt** $4.95 \times 10^{-6}$ or **awrt** $0.00222^2$ |
| | B1 | all correct, but allow full credit if no symbol used; allow symbol other than $\bar{X}$ if correctly defined as sample mean, but do not allow $\mu$ |

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## Question 13(c):

| Answer | Marks | Guidance |
|--------|-------|----------|
| **awrt** $0.136$ seen **BC** | B1 | |
| $\bar{X} < 0.136$ only or $\bar{X} \leq 0.136$ only | B1 | **FT** other correctly defined symbol |

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## Question 13(d):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.1316 < 0.136$ or $0.1316$ is in the critical region (must be correct critical region) oe or $p =$ awrt $0.00008 < 0.05$ oe; **NB** $0.0000799$ or $z =$ awrt $-3.78 < -1.645$ oe | M1 | condone $p =$ awrt $0.00007 < 0.05$ oe **NB** $0.0000740$; or $z =$ awrt $-3.79 < -1.645$ oe from use of $\bar{X} \sim N\left(0.14, \frac{0.0198^2}{80}\right)$ |
| reject $H_0$ | A1 | allow accept $H_1$ or result is significant |
| there is sufficient evidence at the 5% level to **suggest** that the **mean** mass of the apples is **less than** $0.14$ kg | A1 | allow weight; do not allow eg conclude/prove/indicate or other assertive statement instead of suggest |

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