| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Two-tailed hypothesis test |
| Difficulty | Moderate -0.3 This is a straightforward hypothesis testing question with standard binomial distribution work. Part (a) is trivial recall of binomial theorem. Parts (b) and (c) require standard one-tailed test setup and finding critical values from tables, which are routine A-level procedures with no novel problem-solving required. Slightly easier than average due to the straightforward structure. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n2.04b Binomial distribution: as model B(n,p)2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(n = 50\); \(a = \frac{5}{6}, b = \frac{1}{6}\); or \(a = \frac{1}{6}, b = \frac{5}{6}\) | B1, [1] | or \(\left(\frac{5}{6}+\frac{1}{6}\right)^{50}\) or \(\left(\frac{1}{6}+\frac{5}{6}\right)^{50}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = \frac{1}{6}\), allow \(0.167\) or \(0.17\) | B1 | One error, e.g. undefined \(p\) or 2-tail or \(H_1: p > \frac{1}{6}\): B1B0 |
| \(H_1: p < \frac{1}{6}\) | B1 | Two errors, e.g. \(H_1: p > \frac{1}{6}\) and undefined \(p\): B0B0 |
| \(p =\) P(dice shows a 2 on any throw) or \(p =\) proportion of 2s thrown | [2] | Cannot be scored in part (c) |
| Answer | Marks | Guidance |
|---|---|---|
| \(B\!\left(50, \frac{1}{6}\right)\) | B1 | stated or implied |
| Attempt \(P(X \leq a)\) for \(1 \leq a \leq 20\) | M1 | |
| \(P(X \leq 4) = 0.0643\) | or \(P(X < 5) = 0.0643\) | |
| \(P(X \leq 3) = 0.0238\) | A1 | or \(P(X < 4) = 0.0238\). Must see both probabilities correct |
| Rejection region is \(\leq 3\) twos or \(< 4\) twos | A1, [4] | oe. Allow \(X \leq 3\) or \(X < 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(B\!\left(50, \frac{1}{6}\right)\) | B1 | stated or implied |
| Attempt \(P(X \geq a)\) for \(1 \leq a \leq 20\) | M1 | |
| \(P(X \geq 14) = 0.0307\) | A1 | Must see both probabilities correct |
| \(P(X \geq 13) = 0.0627\) | ||
| Rejection region is \(\geq 14\) twos or \(> 13\) twos | A1, [4] | oe. Allow \(X \geq 14\) or \(X > 13\) |
# Question 12:
## Part (a):
$n = 50$; $a = \frac{5}{6}, b = \frac{1}{6}$; or $a = \frac{1}{6}, b = \frac{5}{6}$ | B1, [1] | or $\left(\frac{5}{6}+\frac{1}{6}\right)^{50}$ or $\left(\frac{1}{6}+\frac{5}{6}\right)^{50}$
## Part (b):
$H_0: p = \frac{1}{6}$, allow $0.167$ or $0.17$ | B1 | One error, e.g. undefined $p$ or 2-tail or $H_1: p > \frac{1}{6}$: B1B0
$H_1: p < \frac{1}{6}$ | B1 | Two errors, e.g. $H_1: p > \frac{1}{6}$ and undefined $p$: B0B0
$p =$ P(dice shows a 2 on any throw) or $p =$ proportion of 2s thrown | [2] | Cannot be scored in part (c)
## Part (c):
$B\!\left(50, \frac{1}{6}\right)$ | B1 | stated or implied
Attempt $P(X \leq a)$ for $1 \leq a \leq 20$ | M1 |
$P(X \leq 4) = 0.0643$ | | or $P(X < 5) = 0.0643$
$P(X \leq 3) = 0.0238$ | A1 | or $P(X < 4) = 0.0238$. Must see both probabilities correct
Rejection region is $\leq 3$ twos or $< 4$ twos | A1, [4] | oe. Allow $X \leq 3$ or $X < 4$
**Alternative method if 12(b) $H_1: p > \frac{1}{6}$:**
$B\!\left(50, \frac{1}{6}\right)$ | B1 | stated or implied
Attempt $P(X \geq a)$ for $1 \leq a \leq 20$ | M1 |
$P(X \geq 14) = 0.0307$ | A1 | Must see both probabilities correct
$P(X \geq 13) = 0.0627$ | |
Rejection region is $\geq 14$ twos or $> 13$ twos | A1, [4] | oe. Allow $X \geq 14$ or $X > 13$
---12 The variable $X$ has the distribution $\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)$. The probabilities $\mathrm { P } ( X = r )$ for $r = 0$ to 50 are given by the terms of the expansion of $( a + b ) ^ { n }$ for specific values of $a , b$ and $n$.
\begin{enumerate}[label=(\alph*)]
\item State the values of $a$, $b$ and $n$.
A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the $5 \%$ significance level.
\item Write down suitable hypotheses for the test.
\item Determine the rejection region for the test, showing the values of any relevant probabilities.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q12 [7]}}