| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question requiring routine application of binomial distribution tables to find critical regions and interpret results. While it involves multiple parts and careful probability calculations, it follows a well-practiced procedure with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (i) \(H_0: p = 0.35\); \(H_1: p \neq 0.35\) | B1 | |
| (ii) \(B(15, 0.35)\) | M1 | |
| CR: \(X \leq 1 \cup X \geq 10\) (Allow any letter) | A1A1 | (4) |
| 8 is not in CR | M1 | |
| There is evidence that the Company's claim is true | A1ft | (2) |
| \(0.0142 + 0.0124 = 0.0266\) | B1 | (1) |
| [7] |
| Answer | Marks |
|---|---|
| Item | Guidance |
| (a)(i) | B1: both hypotheses correct. Must mention \(p\) (or \(\pi\)). Words only is B0 |
| (a)(ii) | M1: Writing \(B(15, 0.35)\). May be implied by e.g. \(P(X \leq 1) = 0.0142\) or \(P(X \leq 9) = 0.9876\) |
| 1st A1: \(X < 1\) (accept \(X \leq 2\)). Allow \(0 \leq X \leq 1\) but \(P(X \leq 1)\) is A0 | |
| 2nd A1: \(X > 10\) (accept \(X > 9\)). Allow \(10 \leq X \leq 15\) but \(P(X \geq 10)\) is A0 | |
| Either correct answer will imply M1 | |
| (b) | M1: for a reason that matches their CR. "Interpret" their CR of \(P(X \geq 10)\) as \(X \geq 10\) etc |
| Allow calculation of \(P(X \geq 8) = 1 - 0.8868 = 0.1132\) and "not sig" comment | |
| Do not allow contradictory remarks e.g. 8 is not in CR so significant (this gets M0) | |
| A1ft: for a conclusion correct for their CR in context | |
| Must mention "claim" or "peas" and "germinating" | |
| NB: A correct contextual claim on its own scores M1A1 | |
| (c) | B1: for 0.0266 or awrt 0.0266 (calc gives 0.02662196...) |
| Answer/Working | Marks | Guidance |
|---|---|---|
| (i) $H_0: p = 0.35$; $H_1: p \neq 0.35$ | B1 | |
| (ii) $B(15, 0.35)$ | M1 | |
| | CR: $X \leq 1 \cup X \geq 10$ (Allow any letter) | A1A1 | (4) |
| 8 is not in CR | M1 | |
| There is evidence that the Company's claim is true | A1ft | (2) |
| $0.0142 + 0.0124 = 0.0266$ | B1 | (1) |
| | [7] | |
**Notes:**
| Item | Guidance |
|---|---|
| (a)(i) | B1: both hypotheses correct. Must mention $p$ (or $\pi$). Words only is B0 |
| (a)(ii) | M1: Writing $B(15, 0.35)$. May be implied by e.g. $P(X \leq 1) = 0.0142$ or $P(X \leq 9) = 0.9876$ |
| | 1st A1: $X < 1$ (accept $X \leq 2$). Allow $0 \leq X \leq 1$ but $P(X \leq 1)$ is A0 |
| | 2nd A1: $X > 10$ (accept $X > 9$). Allow $10 \leq X \leq 15$ but $P(X \geq 10)$ is A0 |
| | | Either correct answer will imply M1 |
| (b) | M1: for a reason that matches their CR. "Interpret" their CR of $P(X \geq 10)$ as $X \geq 10$ etc |
| | | Allow calculation of $P(X \geq 8) = 1 - 0.8868 = 0.1132$ and "not sig" comment |
| | | Do not allow contradictory remarks e.g. 8 is not in CR so significant (this gets M0) |
| | A1ft: for a conclusion correct for their CR in context |
| | | Must mention "claim" or "peas" and "germinating" |
| | NB: A correct contextual claim on its own scores M1A1 |
| (c) | B1: for 0.0266 or awrt 0.0266 (calc gives 0.02662196...) |
---5. A company claims that $35 \%$ of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item State suitable hypotheses for a two-tailed test of this claim.
\item Using a $5 \%$ level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to $2.5 \%$ as possible.
\end{enumerate}\item Ann found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer.
\item State the actual significance level of this test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2014 Q5 [7]}}