| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Two-tailed hypothesis test |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis test using normal approximation to binomial with continuity correction. Part (a) requires setting up hypotheses, checking conditions (np and nq > 5), applying continuity correction, and comparing to critical values. Part (b) simply changes to a one-tailed test. The calculations are routine and the question structure is typical of S2 exam questions, making it slightly easier than average overall. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a) No. of 6's \(\sim B(60, p) \approx \text{Po}(60p)\); \(H_0: p = \frac{1}{6}\); \(H_1: p \neq \frac{1}{6}\) | B1 B1 | |
| Under \(H_0\), \(P(X \geq 16 \text{ or } X \leq 4) = 0.0487 + 0.0293 = 0.078 > 5\%\) | M1 A1 A1 | |
| Do not reject \(H_0\) at 5% significance level; accept that \(p = \frac{1}{6}\) | A1 | |
| (b) Now \(H_1: p > \frac{1}{6}\); \(P(X \geq 16) = 0.0487 < 5\%\), so reject \(H_0\) | B1 M1 A1 A1 | Total: 10 marks |
(a) No. of 6's $\sim B(60, p) \approx \text{Po}(60p)$; $H_0: p = \frac{1}{6}$; $H_1: p \neq \frac{1}{6}$ | B1 B1 |
Under $H_0$, $P(X \geq 16 \text{ or } X \leq 4) = 0.0487 + 0.0293 = 0.078 > 5\%$ | M1 A1 A1 |
Do not reject $H_0$ at 5% significance level; accept that $p = \frac{1}{6}$ | A1 |
(b) Now $H_1: p > \frac{1}{6}$; $P(X \geq 16) = 0.0487 < 5\%$, so reject $H_0$ | B1 M1 A1 A1 | Total: 10 marks3. A die is rolled 60 times, and results in 16 sixes.
\begin{enumerate}[label=(\alph*)]
\item Use a suitable approximation to test, at the $5 \%$ significance level, whether the probability of scoring a six is $\frac { 1 } { 6 }$ or not. State your hypotheses clearly.
\item Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than $\frac { 1 } { 6 }$. Carry out this modified test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [10]}}