| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.8 This is a multi-part binomial hypothesis testing question that progresses from routine calculations (parts a-b) through a standard one-tailed test (part c) to a more challenging reverse problem (part d) requiring iterative solution to find minimum sample size. Part (d) elevates this above a typical S2 question as students must work backwards from significance level to determine n, requiring deeper understanding rather than just applying a procedure. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=5) = {}^{20}C_5(0.3)^5(0.7)^{15}\) or \(0.4164 - 0.2375 = 0.17886...\) | M1, A1 | awrt 0.179 |
| Answer | Marks | Guidance |
|---|---|---|
| Mean \(= 6\) | B1 | — |
| \(\text{sd} = \sqrt{20\times0.7\times0.3} = 2.049...\) | M1, A1 | M1 use of \(20\times0.7\times0.3\); awrt 2.05 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p=0.3\), \(H_1: p>0.3\) | B1 | Both hypotheses correct (\(p\) or \(\pi\)) |
| \(X \sim B(20, 0.3)\); \(P(X \geq 8) = 0.2277\) or \(P(X \geq 10) = 0.0480\), so CR \(X \geq 10\) | M1, A1 | M1 using \(B(20,0.3)\); A1 awrt 0.228 or CR \(X \geq 10\) |
| Insufficient evidence to reject \(H_0\) or Not Significant or 8 does not lie in the critical region | dM1 | Correct comment dependent on previous M1 |
| There is no evidence to support the Director (of Studies') belief/There is no evidence that the proportion of parents that do not support the new curriculum is greater than 30% | A1cso | Correct contextual conclusion with underlined words; all previous marks required |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim B(2n, 0.25)\); \(B(8,0.25)\): \(P(X \geq 4)=0.1138\); \(B(10,0.25)\): \(P(X \geq 5)=0.0781\) | M1 | For 0.1138 or 0.0781 or 0.8862 or 0.9219 seen |
| \(2n=10 \Rightarrow n=5\) | A1, A1 | \(B(10,0.25)\) selected; \(n=5\). An answer of 5 with no incorrect working scores 3/3 |
## Question 6:
### Part (a):
| $P(X=5) = {}^{20}C_5(0.3)^5(0.7)^{15}$ or $0.4164 - 0.2375 = 0.17886...$ | M1, A1 | awrt 0.179 |
### Part (b):
| Mean $= 6$ | B1 | — |
| $\text{sd} = \sqrt{20\times0.7\times0.3} = 2.049...$ | M1, A1 | M1 use of $20\times0.7\times0.3$; awrt 2.05 |
### Part (c):
| $H_0: p=0.3$, $H_1: p>0.3$ | B1 | Both hypotheses correct ($p$ or $\pi$) |
| $X \sim B(20, 0.3)$; $P(X \geq 8) = 0.2277$ or $P(X \geq 10) = 0.0480$, so CR $X \geq 10$ | M1, A1 | M1 using $B(20,0.3)$; A1 awrt 0.228 **or** CR $X \geq 10$ |
| Insufficient evidence to reject $H_0$ **or** Not Significant **or** 8 does not lie in the critical region | dM1 | Correct comment dependent on previous M1 |
| There is no evidence to support the Director (of Studies') belief/There is no evidence that the proportion of parents that do not support the new curriculum is greater than 30% | A1cso | Correct contextual conclusion with underlined words; all previous marks required |
### Part (d):
| $X \sim B(2n, 0.25)$; $B(8,0.25)$: $P(X \geq 4)=0.1138$; $B(10,0.25)$: $P(X \geq 5)=0.0781$ | M1 | For 0.1138 or 0.0781 or 0.8862 or 0.9219 seen |
| $2n=10 \Rightarrow n=5$ | A1, A1 | $B(10,0.25)$ selected; $n=5$. An answer of 5 with no incorrect working scores 3/3 |6. The Headteacher of a school claims that $30 \%$ of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, $X$, who do not support the new curriculum is recorded.
Assuming that the Headteacher's claim is correct, find
\begin{enumerate}[label=(\alph*)]
\item the probability that $X = 5$
\item the mean and the standard deviation of $X$
The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than $30 \%$. Given that in the survey of 20 parents 8 do not support the new curriculum,
\item test, at the $5 \%$ level of significance, the Director of Studies' belief. State your hypotheses clearly.
The teachers believe that the sample in the original survey was biased and claim that only $25 \%$ of the parents are in support of the new curriculum. A second random sample, of size $2 n$, is taken and exactly half of this sample supports the new curriculum.
A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size $2 n$
Using the hypotheses $\mathrm { H } _ { 0 } : p = 0.25$ and $\mathrm { H } _ { 1 } : p > 0.25$
\item find the minimum value of $n$ for which the outcome of the test is that the teachers' belief is rejected.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2015 Q6 [13]}}