| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 10 |
| 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 standard S2 binomial distribution question with normal approximation. Parts (a) and (b) require straightforward binomial probability calculations with n=20, p=0.35. Part (c) involves recognizing when to use normal approximation (n=100, np=35, nq=65 both >5) and applying continuity correction. While it requires multiple techniques, these are all routine applications of textbook methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks |
|---|---|
| (a) \(X \sim B(20, 0.35)\) | M1 A1 |
| From tables, \(P(X \leq 4) = 0.118\) | M1 A1 |
| (b) \(P(X = 8) - P(X \leq 7) = 0.7624 - 0.6010 = 0.161\) | M1 A1 |
| (c) \(B(100, 0.35) \approx N(35, 22.75)\) | M1 A1 M1 |
| \(P(X < 25) = P(X < 24.5) = P\left(Z < \frac{-10.5}{4.77}\right) = P(Z < -2.201) = 1 - 0.9861 = 0.0139\) | M1 A1 A1 |
| 10 |
(a) $X \sim B(20, 0.35)$ | M1 A1 |
From tables, $P(X \leq 4) = 0.118$ | M1 A1 |
(b) $P(X = 8) - P(X \leq 7) = 0.7624 - 0.6010 = 0.161$ | M1 A1 |
(c) $B(100, 0.35) \approx N(35, 22.75)$ | M1 A1 M1 |
$P(X < 25) = P(X < 24.5) = P\left(Z < \frac{-10.5}{4.77}\right) = P(Z < -2.201) = 1 - 0.9861 = 0.0139$ | M1 A1 A1 |
| | 10 |On average, 35\% of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that
\begin{enumerate}[label=(\alph*)]
\item less than 5 get A or B grades, [2 marks]
\item exactly 8 get A or B grades. [2 marks]
\end{enumerate}
Five such classes of 20 students are combined to sit the exam.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades. [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [10]}}