| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Session | Specimen |
| Marks | 12 |
| 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 covering routine binomial test procedures. Part (a) requires a straightforward one-tailed binomial test with clear hypotheses and probability calculation. Parts (b) and (c) involve normal approximation to binomial with continuity correction to find critical regions—a textbook application. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation2.05a 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 |
|---|---|---|
| \(H_0: p = 0.30\) | \(H_1: p < 30\) | B1 B1 |
| \(X =\) number ordering vegetarian meal | \(X \sim B(20, 0.30)\) under \(H_0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\therefore\) Not significant i.e. no reason to suspect proportion is lower | M1 A1 A1 ft | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.10\) | \(H_1: p \neq 0.10\) | B1 B1 |
| \(Y =\) number ordering vegetarian meal | \(Y \sim B(100, 0.10) \Rightarrow Y \sim P_0(10)\) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow P(Y \geq 17) = 0.0270\) | M1 A1 | |
| \(\therefore Y \leq 4\) and \(Y \geq 17\) | A1 | (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Significance level is \(0.0270 + 0.0293 = 0.0563\) (5.6%) | B1 ft | (1 mark) |
## Part (a)
$H_0: p = 0.30$ | $H_1: p < 30$ | B1 B1
$X =$ number ordering vegetarian meal | $X \sim B(20, 0.30)$ under $H_0$
$P(X \leq 3) = 0.1071 > 5\%$
$\therefore$ Not significant i.e. no reason to suspect proportion is lower | M1 A1 A1 ft | (5 marks)
## Part (b)
$H_0: p = 0.10$ | $H_1: p \neq 0.10$ | B1 B1
$Y =$ number ordering vegetarian meal | $Y \sim B(100, 0.10) \Rightarrow Y \sim P_0(10)$ | M1
Need $a, b$ such that $P(Y \leq a) \approx 0.025$ and $P(Y \geq b) \approx 0.025$
From tables: $P(Y \leq 4) = 0.0293$ and $P(Y \leq 16) = 0.9730$
$\Rightarrow P(Y \geq 17) = 0.0270$ | M1 A1
$\therefore Y \leq 4$ and $Y \geq 17$ | A1 | (6 marks)
## Part (c)
Significance level is $0.0270 + 0.0293 = 0.0563$ (5.6%) | B1 ft | (1 mark)
**Total: 12 marks**
---In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]
\end{enumerate}
Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
\item State the significance level of this test giving your answer to 2 significant figures. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [12]}}