| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Exact binomial then normal approximation (same context, different n) |
| Difficulty | Standard +0.3 This is a straightforward two-part hypothesis testing question using the binomial distribution. Part (a) requires a standard one-tailed binomial test with clear hypotheses, and part (b) applies the normal approximation to the binomial with continuity correction. Both are routine S2 techniques 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 | Guidance |
|---|---|---|
| (a) If Oldies correct, take \(H_0\): \(P(\text{Trendie}) = 0.45\). Assuming this, distribution of Trendies is \(X \sim N(20, 0.45)\) | B1, M1, A1 | |
| Then \(P(X \geq 11) = 1 - \Phi\sqrt{7} = 0.249 > 5\%\), so do not reject \(H_0\) | M1, A1, A1 | |
| (b) No. of Trendies is \(B(200, 0.55) = N(110, 49.5)\) | B1, M1, A1 | |
| so \(P(X < 100) = P(X < 99.5) = P(Z \leq -10.5/\sqrt{49.5})\) | M1, A1 | |
| \(= P(Z \leq -1.49) = 0.0681\) | A1 | 14 marks total |
(a) If Oldies correct, take $H_0$: $P(\text{Trendie}) = 0.45$. Assuming this, distribution of Trendies is $X \sim N(20, 0.45)$ | B1, M1, A1 |
Then $P(X \geq 11) = 1 - \Phi\sqrt{7} = 0.249 > 5\%$, so do not reject $H_0$ | M1, A1, A1 |
(b) No. of Trendies is $B(200, 0.55) = N(110, 49.5)$ | B1, M1, A1 |
so $P(X < 100) = P(X < 99.5) = P(Z \leq -10.5/\sqrt{49.5})$ | M1, A1 |
$= P(Z \leq -1.49) = 0.0681$ | A1 | 14 marks total |A small opinion poll shows that the Trendies have a $10\%$ lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a $10\%$ lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.
\begin{enumerate}[label=(\alph*)]
\item At the $5\%$ significance level, test which is right, stating your null hypothesis carefully. [6 marks]
\item If it is indeed true that the Trendies are supported by $55\%$ of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes. [8 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [14]}}