| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 12 |
| 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 +0.3 This is a standard S2 hypothesis testing question covering binomial tests and normal approximation. Part (a) is routine binomial hypothesis testing, part (b) tests understanding of assumptions, part (c) applies normal approximation to binomial (a core S2 skill), and part (d) requires basic interpretation. All techniques are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion |
| Answer | Marks |
|---|---|
| (a) \(X \sim B(10, p)\); \(H_0: p = 0.05\), \(H_1: p > 0.05\) | B1 B1 |
| Under \(H_0\), \(P(X \geq 2) = 1 - 0.9139 = 0.0861 > 1\%\), so accept \(H_0\) | M1 A1 A1 |
| (b) Assumed that apples are selected randomly | B1 |
| (c) Now have \(B(60, 0.05)\), assuming \(H_0\). This is approx. \(\text{Po}(3)\) | M1 A1 |
| \(P(X \geq 10) = 1 - 0.9989 = 0.0011 < 1\%\), so reject \(H_0\) | M1 A1 A1 |
| (d) More data gives greater evidence and can be more decisive | B1 |
| 12 |
(a) $X \sim B(10, p)$; $H_0: p = 0.05$, $H_1: p > 0.05$ | B1 B1 |
Under $H_0$, $P(X \geq 2) = 1 - 0.9139 = 0.0861 > 1\%$, so accept $H_0$ | M1 A1 A1 |
(b) Assumed that apples are selected randomly | B1 |
(c) Now have $B(60, 0.05)$, assuming $H_0$. This is approx. $\text{Po}(3)$ | M1 A1 |
$P(X \geq 10) = 1 - 0.9989 = 0.0011 < 1\%$, so reject $H_0$ | M1 A1 A1 |
(d) More data gives greater evidence and can be more decisive | B1 |
| | 12 |A greengrocer sells apples from a barrel in his shop. He claims that no more than 5\% of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly, test his claim at the 1\% significance level. [5 marks]
\item State an assumption that has been made about the selection of the apples. [1 mark]
\item When five other customers also buy 10 apples each, the numbers of bad apples they get are 1, 3, 1, 2 and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the 1\% significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than 5\%. [5 marks]
\item Comment briefly on your results in parts (a) and (c). [1 mark]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [12]}}