| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Multiple binomial probability calculations |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial distribution and hypothesis testing with clear parameters given. Part (a) requires standard binomial probability calculations using B(35, 0.08), and part (b) is a routine one-tailed hypothesis test at 10% significance. All steps are procedural with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| 2(a)(i) \([D = \text{number of bags that are damp}]\) \(D \sim B(35, 0.08)\) NB \(0.08 = \frac{2}{25}\) | M1 | AO 3.3 |
| \(P(D = 2) = 0.2430497\ldots\) awrt 0.243 | A1 | AO 3.4 |
| 2(a)(ii) \(P(D > 3) = [1 - P(D_{\leq} 3) = 1 - 0.69397...] = 0.30602\ldots\) awrt 0.306 | A1 | AO 1.1b |
| 2(b) \(H_0 : p = 0.08\) \(H_1 : p < 0.08\) | B1 | AO 2.5 |
| \([X \sim] B(70, 0.08)\) | M1 | AO 2.1 |
| \([P(X \leq 2)] = 0.0739756\ldots\) awrt 0.074 | A1 | AO 1.1b |
| \([0.074 < 0.10 \text{ so significant, reject } H_0 \text{ so...}]\) there is evidence to support supplier \(B\)'s claim (o.e.) | A1 | AO 2.2b |
**2(a)(i)** $[D = \text{number of bags that are damp}]$ $D \sim B(35, 0.08)$ NB $0.08 = \frac{2}{25}$ | M1 | AO 3.3
$P(D = 2) = 0.2430497\ldots$ awrt **0.243** | A1 | AO 3.4
**2(a)(ii)** $P(D > 3) = [1 - P(D_{\leq} 3) = 1 - 0.69397...] = 0.30602\ldots$ awrt **0.306** | A1 | AO 1.1b
**2(b)** $H_0 : p = 0.08$ $H_1 : p < 0.08$ | B1 | AO 2.5
$[X \sim] B(70, 0.08)$ | M1 | AO 2.1
$[P(X \leq 2)] = 0.0739756\ldots$ awrt **0.074** | A1 | AO 1.1b
$[0.074 < 0.10 \text{ so significant, reject } H_0 \text{ so...}]$ there is evidence to **support supplier $B$'s claim** (o.e.) | A1 | AO 2.2b
**Total: 7 marks**
**Notes:**
**(a)** M1 for selecting a correct model: sight of or use of $B(35, 0.08)$ [Condone $B(0.08, 35)$]. May be implied by one correct answer or sight of $P(D_{\leq} 3) = $ awrt 0.694 (or allow 0.693).
Or seeing $\binom{35}{2} 0.08^2 \times (1-0.08)^{35-2}$
Saying $B(35, 8\%)$ without a correct calculation would score M0.
**(i)** 1st A1 for awrt 0.243
**(ii)** 2nd A1 for awrt 0.306 (Condone poor use of notation e.g. $P(D = 3) = 0.306\ldots$ i.e. just mark ans). NB $P(D...) = 0.539$ scores 2nd A0 but would of course score M1.
**(b)** B1 for both hypotheses correct in terms of $p$ or $\pi$ [Condone 8% for 0.08]
M1 for sight or correct use of $B(70, 0.08)$ [Condone $B(0.08, 70)$]. May be implied by prob of 0.074 or better.
1st A1 for final answer awrt 0.074 can condone poor notation e.g. $P(X_{\leq} 2) = $ awrt 0.074. Can allow this mark for CR of $X_{\leq} 2$ provided $[P(X_{\leq} 2)] = 0.074$ (or better) is seen [Can allow 0.07 if $X - B(70, 0.08)$ and $P(X_{\leq} 2)$ are both seen]
2nd A1 (dep on M1A1 but independent of hypotheses) for a correct inference in context. Must mention **claim** or **$B$** and idea of **support** for ... or proportion/probability (of damp bags) and idea of **less than 8%** or A 2nd A0 for contradictory statements e.g. "accept $H_0$ so evidence to support $B$'s claim". 2nd A0 if you see $0.0739... < 0.08$ so significant/ reject $H_0$ etc.
**MR** 0.8 for 0.08: In (a) allow M1 for $B(35, 0.8)$ then A0A0. In (b) allow B1 for Hypotheses and M1 for $B(70, 0.8)$ seen, then A0A0.
---\begin{enumerate}
\item A manufacturer of sweets knows that $8 \%$ of the bags of sugar delivered from supplier $A$ will be damp.\\
A random sample of 35 bags of sugar is taken from supplier $A$.\\
(a) Using a suitable model, find the probability that the number of bags of sugar that are damp is\\
(i) exactly 2\\
(ii) more than 3
\end{enumerate}
Supplier $B$ claims that when it supplies bags of sugar, the proportion of bags that are damp is less than $8 \%$
The manufacturer takes a random sample of 70 bags of sugar from supplier $B$ and finds that only 2 of the bags are damp.\\
(b) Carry out a suitable test to assess supplier B's claim.
You should state your hypotheses clearly and use a $10 \%$ level of significance.
\hfill \mbox{\textit{Edexcel AS Paper 2 2022 Q2 [7]}}