| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | October |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Small sample binomial probability |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard binomial calculations and normal approximation. Part (a) requires basic binomial probability calculations with n=10; part (b) is a routine normal approximation with continuity correction to find critical values; part (c) is a standard one-tailed hypothesis test. All parts follow textbook procedures with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim B(10, 0.45)\) | M1 | Writing or using \(B(10, 0.45)\) in (i) or (ii) implied by correct answer |
| \(P(X \leq 1) = 0.0233\) | A1 | awrt 0.0233 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(X \geq 6) = 1 - P(X \leq 5)\) or \(1 - 0.7384\) | M1 | For writing or using \(1 - P(X \leq 5)\) oe |
| \(= 0.2616...\) | A1 | awrt 0.262 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F \sim N(54, 29.7)\) | M1A1 | M1 for \(N(54,...)\), A1 for \(N(54, 29.7)\) |
| \(\frac{c + 0.5 - 54}{\sqrt{29.7}} \leq -1.6449\) or \(\frac{d - 0.5 - 54}{\sqrt{29.7}} \geq 1.6449\) | M1M1B1 | M1 standardising (allow \(\pm\)) using "54" and "29.7", \(1 < |
| \(c = 44\) and \(d = 64\) | A1cso | One correct inequality A1; both \(c\) and \(d\) correct integers A1cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.45\), \(H_1: p < 0.45\) | B1 | Both hypotheses correct in terms of \(p\) or \(\pi\), must be attached to \(H_0\) and \(H_1\) |
| \(Y \sim B(30, 0.45)\), \(P(Y \leq 8) = 0.03...\) or CR \(Y \leq 8\) | B1 | 0.03 or better (0.03120...) or CR stated as \(Y \leq 8\) |
| 8 is in the critical region or Reject \(H_0\) | dM1 | Dep on 2nd B1; correct statement, allow opposite conclusion if 2-tail hypotheses given |
| Therefore the data supports the manufacturer's claim | A1 | Correct conclusion for their \(H_1\); allow proportion/percentage/probability/number/amount of flawed plates has decreased/reduced/is not 0.45/has changed oe |
# Question 3:
## Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim B(10, 0.45)$ | M1 | Writing or using $B(10, 0.45)$ in (i) or (ii) implied by correct answer |
| $P(X \leq 1) = 0.0233$ | A1 | awrt 0.0233 |
## Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X \geq 6) = 1 - P(X \leq 5)$ or $1 - 0.7384$ | M1 | For writing or using $1 - P(X \leq 5)$ oe |
| $= 0.2616...$ | A1 | awrt 0.262 |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F \sim N(54, 29.7)$ | M1A1 | M1 for $N(54,...)$, A1 for $N(54, 29.7)$ |
| $\frac{c + 0.5 - 54}{\sqrt{29.7}} \leq -1.6449$ **or** $\frac{d - 0.5 - 54}{\sqrt{29.7}} \geq 1.6449$ | M1M1B1 | M1 standardising (allow $\pm$) using "54" and "29.7", $1 < |z| < 2$, condone missing $\pm 0.5$; M1 for continuity correction $\pm 0.5$; B1 for 1.6449 or better |
| $c = 44$ and $d = 64$ | A1cso | One correct inequality A1; both $c$ and $d$ correct integers A1cso |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.45$, $H_1: p < 0.45$ | B1 | Both hypotheses correct in terms of $p$ or $\pi$, must be attached to $H_0$ and $H_1$ |
| $Y \sim B(30, 0.45)$, $P(Y \leq 8) = 0.03...$ or CR $Y \leq 8$ | B1 | 0.03 or better (0.03120...) or CR stated as $Y \leq 8$ |
| 8 is in the critical region or Reject $H_0$ | dM1 | Dep on 2nd B1; correct statement, allow opposite conclusion if 2-tail hypotheses given |
| Therefore the data supports the **manufacturer's claim** | A1 | Correct conclusion for their $H_1$; allow **proportion/percentage/probability/number/amount** of flawed **plates** has **decreased/reduced/is not 0.45/has changed** oe |
---3. A manufacturer produces plates. The proportion of plates that are flawed is $45 \%$, with flawed plates occurring independently.
A random sample of 10 of these plates is selected.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the sample contains
\begin{enumerate}[label=(\roman*)]
\item fewer than 2 flawed plates,
\item at least 6 flawed plates.\\
(4)
George believes that the proportion of flawed plates is not $45 \%$. To assess his belief George takes a random sample of 120 plates. The random variable $F$ represents the number of flawed plates found in the sample.
\end{enumerate}\item Using a normal approximation, find the maximum number of plates, $c$, and the minimum number of plates, $d$, such that
$$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$
where $F \sim \mathrm {~B} ( 120,0.45 )$
The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
\item Use a suitable hypothesis test, at the $5 \%$ level of significance, to assess the manufacturer's claim. State your hypotheses clearly.
\includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-11_2255_50_314_34}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2020 Q3 [15]}}