| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| 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 binomial table lookup and understanding of two-tailed tests. While it involves multiple steps (finding critical values in both tails, calculating actual significance level), the procedure is routine and follows textbook methods with no conceptual challenges beyond applying the definition of critical regions. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(X \sim B(25, 0.5)\) | M1 | May be implied by calculations in part a or b |
| \(P(X \leq 7) = 0.0216\) | Note: just seeing either \(P(X\leq 7)\) or \(P(X\geq 18)\) scores M1 A0 A0 | |
| \(P(X \geq 18) = 0.0216\) | ||
| CR: \(X \leq 7\) \(\cup\) \(X \geq 18\) | A1, A1 | 1st A1: also allow \(X<8\) or \([0,7]\) or \(0\leq X\leq 7\) oe. DO NOT allow \(P(X\leq 7)\) or \(7-0\). 2nd A1: also allow \(X>17\) or \([18,25]\) oe. DO NOT allow \(P(X\geq 18)\) or \(18-25\). SC: \(7\geq X\geq 18\) gains M1 A1 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(\text{rejecting } H_0) = 0.0216 + 0.0216\) | M1 | Adding the two critical regions' probabilities; may be awarded for awrt \(0.0432\) |
| \(= 0.0432\) | A1 | awrt \(0.0432\)/\(0.0433\) |
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $X \sim B(25, 0.5)$ | M1 | May be implied by calculations in part a or b |
| $P(X \leq 7) = 0.0216$ | | Note: just seeing either $P(X\leq 7)$ or $P(X\geq 18)$ scores M1 A0 A0 |
| $P(X \geq 18) = 0.0216$ | | |
| CR: $X \leq 7$ $\cup$ $X \geq 18$ | A1, A1 | 1st A1: also allow $X<8$ or $[0,7]$ or $0\leq X\leq 7$ oe. **DO NOT** allow $P(X\leq 7)$ or $7-0$. 2nd A1: also allow $X>17$ or $[18,25]$ oe. **DO NOT** allow $P(X\geq 18)$ or $18-25$. SC: $7\geq X\geq 18$ gains M1 A1 A0 |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(\text{rejecting } H_0) = 0.0216 + 0.0216$ | M1 | Adding the two critical regions' probabilities; may be awarded for awrt $0.0432$ |
| $= 0.0432$ | A1 | awrt $0.0432$/$0.0433$ |2. A test statistic has a distribution $\mathrm { B } ( 25 , p )$.
Given that
$$\mathrm { H } _ { 0 } : p = 0.5 \quad \mathrm { H } _ { 1 } : p \neq 0.5$$
\begin{enumerate}[label=(\alph*)]
\item find the critical region for the test statistic such that the probability in each tail is as close as possible to $2.5 \%$.
\item State the probability of incorrectly rejecting $\mathrm { H } _ { 0 }$ using this critical region.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2012 Q2 [5]}}