| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Multiple binomial probability calculations |
| Difficulty | Standard +0.3 This is a straightforward multi-part binomial question requiring standard calculations: (a) P(X≥3) for B(9,1/6), (b) combining two binomial distributions, and (c) a routine one-tailed hypothesis test. All parts follow textbook procedures with no novel insight required, though part (b) requires careful thinking about the compound probability structure. Slightly easier than average due to clear setup and standard techniques. |
| 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Let \(C\) = number of successful calls. \(C \sim B\!\left(9, \dfrac{1}{6}\right)\) | M1 | For selecting the right model |
| \(P(C \geq 3) = 1 - P(C \leq 2) = 0.1782\ldots\) awrt \(0.178\) | A1 | awrt \(0.178\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Let \(X\) = number of occasions when at least 3 calls are successful. \(P(X=1) = 5 \times (\text{"0.1782..."}) \times (\text{"0.8217..."})^4\) | M1 | For \(5 \times (\text{"their}(a)\text{"}) \times (1 - \text{"their}(a)\text{"})^4\) |
| \(= 0.4061\ldots\) awrt \(0.406\) | A1 | awrt \(0.406\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: p = \dfrac{1}{6} \qquad H_1: p > \dfrac{1}{6}\) | B1 | For correctly stating both hypotheses in terms of \(p\) or \(\pi\). Accept \(p = 0.1\dot{6}\) |
| Let \(R\) = number of successful calls \(R \sim B\!\left(35, \dfrac{1}{6}\right)\) | M1 | For selecting a suitable model. May be implied by a correct probability or CR |
| \(P(R \geq 11) = 1 - P(R \leq 10) = 0.02\ldots\) | A1 | Correct probability \(0.02\) or better \((0.02318\ldots)\); CR: \(R \geq 11\) and either \(P(R \leq 9) = 0.9450\) or \(P(R \leq 10) = 0.9768\) or \(1 - P(R \leq 10) = 0.0232\) |
| There is sufficient evidence to support that Rowan has more successful sales calls than Afrika. | A1 | Dependent on M1A1 but can ignore hypotheses. Conclusion in context supporting Rowan's belief / Rowan is a better sales person. Do not accept "Rowan can reject \(H_0\)" |
# Question 5:
## Part 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Let $C$ = number of successful calls. $C \sim B\!\left(9, \dfrac{1}{6}\right)$ | M1 | For selecting the right model |
| $P(C \geq 3) = 1 - P(C \leq 2) = 0.1782\ldots$ awrt $0.178$ | A1 | awrt $0.178$ |
## Part 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Let $X$ = number of occasions when at least 3 calls are successful. $P(X=1) = 5 \times (\text{"0.1782..."}) \times (\text{"0.8217..."})^4$ | M1 | For $5 \times (\text{"their}(a)\text{"}) \times (1 - \text{"their}(a)\text{"})^4$ |
| $= 0.4061\ldots$ awrt $0.406$ | A1 | awrt $0.406$ |
## Part 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p = \dfrac{1}{6} \qquad H_1: p > \dfrac{1}{6}$ | B1 | For correctly stating both hypotheses in terms of $p$ or $\pi$. Accept $p = 0.1\dot{6}$ |
| Let $R$ = number of successful calls $R \sim B\!\left(35, \dfrac{1}{6}\right)$ | M1 | For selecting a suitable model. May be implied by a correct probability or CR |
| $P(R \geq 11) = 1 - P(R \leq 10) = 0.02\ldots$ | A1 | Correct probability $0.02$ or better $(0.02318\ldots)$; CR: $R \geq 11$ and either $P(R \leq 9) = 0.9450$ or $P(R \leq 10) = 0.9768$ or $1 - P(R \leq 10) = 0.0232$ |
| There is sufficient evidence to support that **Rowan** has more successful sales calls than Afrika. | A1 | Dependent on M1A1 but can ignore hypotheses. Conclusion in context supporting **Rowan's** belief / Rowan is a better sales person. Do not accept "Rowan can reject $H_0$" |\begin{enumerate}
\item Afrika works in a call centre.
\end{enumerate}
She assumes that calls are independent and knows, from past experience, that on each sales call that she makes there is a probability of $\frac { 1 } { 6 }$ that it is successful.
Afrika makes 9 sales calls.\\
(a) Calculate the probability that at least 3 of these sales calls will be successful.
The probability of Afrika making a successful sales call is the same each day.\\
Afrika makes 9 sales calls on each of 5 different days.\\
(b) Calculate the probability that at least 3 of the sales calls will be successful on exactly 1 of these days.
Rowan works in the same call centre as Afrika and believes he is a more successful salesperson.
To check Rowan's belief, Afrika monitors the next 35 sales calls Rowan makes and finds that 11 of the sales calls are successful.\\
(c) Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence to support Rowan's belief.
\hfill \mbox{\textit{Edexcel AS Paper 2 2020 Q5 [8]}}