| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Moderate -0.3 This is a standard S2 hypothesis testing question requiring routine application of binomial tables to find critical regions and interpret results. While it involves multiple steps (finding two-tailed critical values, calculating actual significance level, and drawing a conclusion), each step follows a well-practiced procedure with no novel problem-solving required. It's slightly easier than average because the mechanics are straightforward once you know the method, though the two-tailed aspect and actual significance level calculation add minor complexity. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim B(20, 0.3)\) | M1 | |
| \(P(X \leq 2) = 0.0355\) | ||
| \(P(X \geq 11) = 1 - 0.9829 = 0.0171\) | ||
| Critical region is \((X \leq 2) \cup (X \geq 11)\) | A1, A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Significance level \(= 0.0355 + 0.0171 = 0.0526\) or 5.26% | M1, A1 | (2 marks) |
| Answer | Marks |
|---|---|
| Insufficient evidence to reject \(H_0\) Or sufficient evidence to accept \(H_0\)/not significant; \(x = 3\) (or the value) is not in the critical region or \(0.1071 > 0.025\) | B1 ft, B1 ft |
| Do not allow inconsistent comments | (2 marks) |
## Part (a)
$X \sim B(20, 0.3)$ | M1 |
$P(X \leq 2) = 0.0355$ |
$P(X \geq 11) = 1 - 0.9829 = 0.0171$ |
Critical region is $(X \leq 2) \cup (X \geq 11)$ | A1, A1 | (3 marks)
## Part (b)
Significance level $= 0.0355 + 0.0171 = 0.0526$ or 5.26% | M1, A1 | (2 marks)
## Part (c)
Insufficient evidence to reject $H_0$ Or sufficient evidence to accept $H_0$/not significant; $x = 3$ (or the value) is not in the critical region or $0.1071 > 0.025$ | B1 ft, B1 ft |
Do not allow inconsistent comments | (2 marks)
---A single observation $x$ is to be taken from a Binomial distribution B(20, $p$).
This observation is used to test $H_0 : p = 0.3$ against $H_1 : p \neq 0.3$
\begin{enumerate}[label=(\alph*)]
\item Using a 5\% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5\%. [3]
\item State the actual significance level of this test. [2]
\end{enumerate}
The actual value of $x$ obtained is 3.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item State a conclusion that can be drawn based on this value giving a reason for your answer. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2009 Q3 [7]}}