| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Expected value and most likely value |
| Difficulty | Standard +0.3 This is a straightforward S2 question combining basic probability distributions with a standard one-tailed hypothesis test. Part (a)-(c) involve routine binomial probability calculations with a simple transformation, while parts (d)-(e) are textbook critical region questions requiring only table lookup and cumulative probability addition—no novel insight or complex reasoning required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(P(D = -1) = [=P(\text{1 blue and 2 red marbles selected})] = 3 \times 0.2 \times 0.8^2 = 0.384*\) | M1 A1 cso (2) |
| (b) | d | |
| 0.8³ | \([3 \times 0.2 \times 0.8^2]\) | \(3 \times 0.2^2 \times 0.8\) |
| P(D = d) | 0.512 = \(\frac{64}{125}\) | \([0.384 = \frac{96}{125}]\) |
| B1 M1 A1 (3) | B1 for all correct \(d\)-values / M1 for correct expression for at least 1 other probability / A1 for a complete distribution | |
| (c) | –3 | B1 (1) |
| (d) | \(X \sim \text{B}(12, 0.2)\) \(P(X \leq 4) = 0.9274\), \(P(X \leq 3) = 0.7946\), \(P(X \geq 5) = 0.0726 < 0.10\) \(P(X \geq 4) = 0.2054 > 0.10\), CR: \(X \geq 5\) | M1 A1 (2) |
| (e) | 0.0726 | B1 ft (1) Total 9 |
(a) | $P(D = -1) = [=P(\text{1 blue and 2 red marbles selected})] = 3 \times 0.2 \times 0.8^2 = 0.384*$ | M1 A1 cso (2) | M1 for identifying 1 blue and 2 red leading to $0.2 \times (1 - 0.2)^2$ / A1 cso for a complete correct calculation $3 \times 0.2 \times 0.8^2$
(b) | | d | –3 | [–1] | 1 | 3 |
| | | 0.8³ | $[3 \times 0.2 \times 0.8^2]$ | $3 \times 0.2^2 \times 0.8$ | 0.2³ |
| | P(D = d) | 0.512 = $\frac{64}{125}$ | $[0.384 = \frac{96}{125}]$ | $0.096 = \frac{15}{123}$ | 0.008 = $\frac{1}{125}$ |
| | | B1 M1 A1 (3) | B1 for all correct $d$-values / M1 for correct expression for at least 1 other probability / A1 for a complete distribution
(c) | –3 | B1 (1) |
(d) | $X \sim \text{B}(12, 0.2)$ $P(X \leq 4) = 0.9274$, $P(X \leq 3) = 0.7946$, $P(X \geq 5) = 0.0726 < 0.10$ $P(X \geq 4) = 0.2054 > 0.10$, CR: $X \geq 5$ | M1 A1 (2) | 1st M1 for using $X \sim \text{B}(12, 0.2)$ to find a relevant probability to determine a critical region / P(X ≥ 5) as final answer is M1A0
(e) | 0.0726 | B1 ft (1) Total 9 | B1 ft for a significance level consistent with their CR from (d). Must come from a one-tailed test from $X \sim \text{B}(12, 0.2)$ so may see $P(X \geq 6) = 0.0194$ or $P(X \geq 7) = 0.0039$
---4. A bag contains a large number of marbles, each of which is blue or red.
A random sample of 3 marbles is taken from the bag.
The random variable $D$ represents the number of blue marbles taken minus the number of red marbles taken.
Given that 20\% of the marbles in the bag are blue,
\begin{enumerate}[label=(\alph*)]
\item show that $\mathrm { P } ( D = - 1 ) = 0.384$
\item find the sampling distribution of $D$
\item write down the mode of $D$
Takashi claims that the true proportion of blue marbles is greater than 20\% and tests his claim by selecting a random sample of 12 marbles from the bag.
\item Find the critical region for this test at the 10\% level of significance.
\item State the actual significance level of this test.
\includegraphics[max width=\textwidth, alt={}, center]{d2f40cdb-917a-4377-88f4-396766a299e2-15_2255_47_314_37}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q4 [9]}}