# Question 5:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.3$ | B1 | Must use $p$ or $\pi$ |
| $H_1: p > 0.3$ | B1 | Must use $p$ and be one tail |
| Let X represent the number of tomatoes greater than 4 cm: $X \sim B(40, 0.3)$ | B1 | May be implied by calculation |
| $P(X \geq 18) = 1 - P(X \leq 17)$ | M1 | Attempt to find $1 - P(X \leq 17)$ or get a correct probability. For CR method must attempt to find $P(X \geq 18)$ or give correct critical region |
| $= 0.0320$ | A1 | awrt 0.032 or correct CR: $X \geq 18$ |
| $0.0320 < 0.05$, so reject $H_0$ / significant | M1 | Correct statement based on probability, $H_1$ and 0.05, or correct contextualised statement |
| New fertiliser has increased the probability of a tomato being greater than 4 cm / Dhriti's claim is true | B1 dep | Must use words: increased, tomato, and reference to size or diameter. Dependent on previous M1 |

**Two tail test would get: B1 B0 B1 M1 A1 M1 B0**

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# Question 6a(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Let $X$ represent the number of sunflower plants more than 1.5m high; $X \sim Po(10)$, $\mu = 10$ | B1 | Mean = 10, may be implied in (i) or (ii) |
| $P(8 \leq X \leq 13) = P(X \leq 13) - P(X \leq 7)$ | M1 | Attempting to find $P(X \leq 13) - P(X \leq 7)$ |
| $= 0.8645 - 0.2202 = 0.6443$ | A1 | awrt 0.644 |

# Question 6a(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim N(10, 7.5)$ | A1 | |
| $\sigma^2 = 7.5$ | B1 | May be implied by correct standardised formula |
| $P(7.5 \leq X \leq 13.5) = P\!\left(\dfrac{7.5-10}{\sqrt{7.5}} \leq X \leq \dfrac{13.5-10}{\sqrt{7.5}}\right)$ | M1 | Using 7.5 or 8.5 or 12.5 or 13.5 |
| $= P(-0.913 \leq X \leq 1.278)$ | M1 | Standardising using correct values and mean and SD. A1 for each z-value: awrt $-0.91$ and awrt $1.28$ |
| $= 0.8997 - (1 - 0.8186)$ | M1 | Finding correct area; prob $> 0.5$ – prob $< 0.5$ or prob $< 0.5$ + prob $< 0.5$ |
| $= 0.7183$ | A1 A1 | awrt 0.718 or 0.719. Dependent on all three method marks |

# Question 6b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Normal approximation (not Poisson) | M1 | |
| Since $n$ is large and $p$ close to half, **or** $np = 10$, $npq = 7.5$ so mean $\neq$ variance, **or** $np (= 10)$ and $nq (= 30)$ both $> 5$ | A1 | Do **not** allow $np > 5$ and $npq > 5$ |
| or exact binomial $= 0.7148$ | B1 dep | |

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# Question 7a(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| A hypothesis test is a mathematical procedure to examine a value of a population parameter proposed by the null hypothesis compared with an alternative hypothesis | B1 | Method for deciding between 2 hypotheses |

# Question 7a(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| The critical region is the range of values or a test statistic or region where the test is significant | B1g | Range of values — may be implied by other words. Not "region" on its own |
| that would lead to the rejection of $H_0$ | B1h | Which leads you to reject $H_0$ |

# Question 7b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Let X represent the number of incoming calls: $X \sim Po(9)$ | B1 | Using $P_o(9)$ |
| $P(X \geq 16) = 0.0220$ | M1 A1 | Attempting to find $P(X \geq 16)$ or $P(x \leq 3)$; A1: 0.0220 or $P(X \geq 16)$ |
| $P(x \leq 3) = 0.0212$ | A1 | 0.0212 or $P(x \leq 3)$ |
| Critical region: $x \leq 3$ or $x \geq 16$ | B1 | Correct critical region. A completely correct CR gets all 5 marks |

# Question 7c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Significance level $= 0.0220 + 0.0212 = 0.0432$ or $4.32\%$ | B1 | awrt 0.0432 |

# Question 7d:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 0.45$; $H_1: \lambda < 0.45$ (accept $H_0: \lambda = 4.5$; $H_1: \lambda < 4.5$) | B1 | May use $\lambda$ or $\mu$. Needs both $H_0$ and $H_1$ |
| Using $X \sim Po(4.5)$ | M1 | Using $P_o(4.5)$ |
| $P(X \leq 1) = 0.0611$; CR: $X \leq 0$ | A1 | Correct probability or CR only |
| $0.0611 > 0.05$; $1 \geq 0$ or 1 not in the critical region | M1 | Correct statement based on probability, $H_1$ and 0.05, or correct contextualised statement |
| There is evidence to accept $H_0$ / not significant | B1 cao | Must see word **calls** in conclusion |
| There is no evidence that there are fewer calls during school holidays | | Conclusion in context |