| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find sample size for test |
| Difficulty | Standard +0.3 This is a straightforward two-tailed hypothesis test using binomial distribution with standard procedures. Part (i) requires basic calculation of P(R≥6) for B(6,0.45) and doubling for two-tailed test. Part (ii) involves using tables to find critical values, which is routine table work. Both parts follow standard S2 methodology with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Find \(P(R \geq 6)\) or \(P(R < 6) = 0.0083\) or \(0.9917\) | M1, A1 | Find \(P(=6)\) from tables/calc, OR RH critical region. \(P(\geq 6)\) in range [0.008, 0.0083] or \(P(< 6) = 0.9917\) |
| Compare with \(0.025\) [can be from N] | B1 | [0.05 if "empty LH tail stated"]. Reject \(H_0\) |
| (ii) \(n = 9\), \(P(\leq 1) = 0.0385\) [\(> 0.025\)]; \(n = 10\), \(P(\leq 1) = 0.0233\) [\(< 0.025\)]. Therefore \(n = 9\) | M1, A1, B1 | OR \(n = 8\), \(P(\leq 1) = 0.0632\). Both of these probabilities seen, don't need 0.025. Answer \(n = 9\) only, indept of M1A1, not from \(P(=1)\) |
(i) Find $P(R \geq 6)$ or $P(R < 6) = 0.0083$ or $0.9917$ | M1, A1 | Find $P(=6)$ from tables/calc, OR RH critical region. $P(\geq 6)$ in range [0.008, 0.0083] or $P(< 6) = 0.9917$
Compare with $0.025$ [can be from N] | B1 | [0.05 if "empty LH tail stated"]. Reject $H_0$
(ii) $n = 9$, $P(\leq 1) = 0.0385$ [$> 0.025$]; $n = 10$, $P(\leq 1) = 0.0233$ [$< 0.025$]. Therefore $n = 9$ | M1, A1, B1 | OR $n = 8$, $P(\leq 1) = 0.0632$. Both of these probabilities seen, don't need 0.025. Answer $n = 9$ only, indept of M1A1, not from $P(=1)$
---2 (i) The random variable $R$ has the distribution $\mathrm { B } ( 6 , p )$. A random observation of $R$ is found to be 6. Carry out a $5 \%$ significance test of the null hypothesis $\mathrm { H } _ { 0 } : p = 0.45$ against the alternative hypothesis $\mathrm { H } _ { 1 } : p \neq 0.45$, showing all necessary details of your calculation.\\
(ii) The random variable $S$ has the distribution $\mathrm { B } ( n , p ) . \mathrm { H } _ { 0 }$ and $\mathrm { H } _ { 1 }$ are as in part (i). A random observation of $S$ is found to be 1 . Use tables to find the largest value of $n$ for which $\mathrm { H } _ { 0 }$ is not rejected. Show the values of any relevant probabilities.
\hfill \mbox{\textit{OCR S2 2006 Q2 [7]}}