# Question 3:

## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| A list of all the **shops** | B1 | For the idea of a list/database of **all shops**; list of all stocktaking systems is B0 |

## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| The shops | B1 | For allowing shop or store(s); the number of shops is B0; the stocktaking systems at each shop is B0 |

## Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Advantage: A sample is quicker/cheaper/easier to process | B1 | For a correct advantage; e.g. allow census takes longer than a sample; 'a sample is more uncertain' on its own is B0 |
| Disadvantage: less accurate/may be biased/may not be representative | B1 | For a correct disadvantage e.g. a census is more accurate than a sample. If no reference to sample or census assume referring to sample. Ignore extraneous non-contradictory comments |

## Part (d):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $P(X \leq 6) = 0.0172$ or $P(X \leq 18) = 0.0212$ or $P(X \geq 17) = 0.9788$ or $X \leq 6$ or $X \geq 18$ | M1 | For one of these probability statements correct or awrt 0.017 or awrt 0.021 or awrt 0.98 or one correct CR |
| $[P(X \leq 6)] = 0.0172$ **and** $[P(X \leq 18)] = 0.0212$ | A1 | For both probabilities awrt 0.0172 and awrt 0.0212 |
| CR: $[0, 6] \cup X \geq 6$, $6$, $18$, $X[\leq 30]$ | A1 | For both CR correct e.g. $X < 7$, $X > 17$; ignore symbol between two CR tails; allow any letter |

## Part (e):
| Working/Answer | Mark | Guidance |
|---|---|---|
| 20 is in the critical region therefore there is evidence that **Jian's belief** is incorrect | B1ft | For stating 20 is in the CR **and** give correct statement. Only ft if CR is in form $X \leq C_1 \cup X \geq C_2$ |

## Part (f):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.4 \quad H_1: p < 0.4$ | B1 | For both hypotheses correct, using $p$ or $\pi$. Must be attached to $H_0$ and $H_1$ |
| $J \sim B(150, 0.4) \Rightarrow \approx N(60, 36)$ | M1A1 | For writing or using $N(60,...)$; for writing or using $N(60, 36)$ |
| $P\left(J \leq 47\right) \approx P\left(Z \leq \frac{47.5 - 60}{6}\right) = -2.08333...$ | M1 | For standardising (allow $\pm$) using "60" and "6" with either 46.5, 47 or 47.5 |
| $\frac{(n+0.5)-60}{6} = -1.6449$ | M1 | For CR method: $n$, $n+0.5$ or $n-0.5$ and equate to $-1.6449$ or better |
| $= 0.0188$ (calc 0.018610...) | A1 | For awrt 0.019 or CR: $J <$ awrt 49.6 or $J + 0.5 <$ awrt 50.1 |
| There is sufficient evidence that the **proportion** of shops using system incorrectly **is less than 0.4/decreased** | A1 | Dep on previous A1 for correct conclusion in context using bold word. Do not allow 'number' for 'proportion' |

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