| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Identify distribution and parameters |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard binomial distribution identification, probability calculations using tables, and a basic one-tailed hypothesis test. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-part structure and hypothesis testing component. |
| 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 variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim Bin(20, 0.4)\) | B1, B1 | Bin, 20 & 0.4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(5 < X < 15) = 0.9984 - 0.1256 = 0.8728\) | M1, M1(dep), A1A1 | \(\leq 14\) & \(\leq 5\), subtract, both correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 20 \times 0.4 = 8\) | B1 | 8 |
| \(sd = \sqrt{20 \times 0.4 \times 0.6} = 2.19\) | M1, A1 | Sub in \(\sqrt{npq}\), 2.19 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.4\) | B1 | Both hypotheses |
| \(H_1: p > 0.4\) | B1 | |
| \(P(X \geq 8 \mid n=10,\, p=0.4) = 1 - 0.9877 = 0.0123\) | M1, A1 | Require '1 minus' |
| Reject \(H_0\) | M1 | |
| Proportion of diners who prefer organic food is higher than trade magazine's claim | A1ft | Context |
| [Note: \(P(X \leq 6) = 0.9452\), \(P(X \leq 7) = 0.9877\)] | M1A1 |
# Question 6:
## Part (a)
| $X \sim Bin(20, 0.4)$ | B1, B1 | Bin, 20 & 0.4 |
## Part (b)
| $P(5 < X < 15) = 0.9984 - 0.1256 = 0.8728$ | M1, M1(dep), A1A1 | $\leq 14$ & $\leq 5$, subtract, both correct |
## Part (c)
| $E(X) = 20 \times 0.4 = 8$ | B1 | 8 |
| $sd = \sqrt{20 \times 0.4 \times 0.6} = 2.19$ | M1, A1 | Sub in $\sqrt{npq}$, 2.19 |
## Part (d)
| $H_0: p = 0.4$ | B1 | Both hypotheses |
| $H_1: p > 0.4$ | B1 | |
| $P(X \geq 8 \mid n=10,\, p=0.4) = 1 - 0.9877 = 0.0123$ | M1, A1 | Require '1 minus' |
| Reject $H_0$ | M1 | |
| Proportion of diners who prefer organic food is higher than trade magazine's claim | A1ft | Context |
| [Note: $P(X \leq 6) = 0.9452$, $P(X \leq 7) = 0.9877$] | M1A1 | |
---6. The owner of a small restaurant decides to change the menu. A trade magazine claims that $40 \%$ of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.
\begin{enumerate}[label=(\alph*)]
\item Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners $X$ who choose organic foods.
\item Find $\mathrm { P } ( 5 < X < 15 )$.
\item Find the mean and standard deviation of $X$.
The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.
\item Test, at the $5 \%$ level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly.\\
(5)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2002 Q6 [14]}}