Standard +0.8 This question requires students to work backwards from a desired significance level to determine sample size and critical region, consulting cumulative binomial tables for multiple values of n. It demands understanding of how significance levels relate to critical regions and involves systematic comparison across different sample sizes—more conceptually demanding than routine hypothesis test execution.
1 A random variable has the distribution \(\mathrm { B } ( n , p )\). It is required to test \(\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }\) against \(\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }\) at a significance level as close to \(1 \%\) as possible, using a sample of size \(n = 8,9\) or 10 . Use tables to find which value of \(n\) gives such a test, stating the critical region for the test and the corresponding significance level. [0pt]
[4]
2 seen but not \(\leq\): M1A0. Allow "\(P(\leq 2)\)". CR must be stated explicitly for A1. SR: \(\leq 3\) with 0.0424: (B1)M1A0. SR: If 0, give B1 for at least 3 of 0.0083, 0.0113, 0.0026, 0.0197, 0.0034 seen
0.0083
A1
Or more SF. "\(n=9\), \(CR \geq 3\)", 0.0083 seen: B1M1A0A1
[4 marks]
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n = 9$ | B1 | Stated explicitly |
| CR is $\leq 2$ | M1A1 | 2 seen but not $\leq$: M1A0. Allow "$P(\leq 2)$". CR must be stated explicitly for A1. SR: $\leq 3$ with 0.0424: (B1)M1A0. SR: If 0, give B1 for at least 3 of 0.0083, 0.0113, 0.0026, 0.0197, 0.0034 seen |
| **0.0083** | A1 | Or more SF. "$n=9$, $CR \geq 3$", 0.0083 seen: B1M1A0A1 |
**[4 marks]**
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1 A random variable has the distribution $\mathrm { B } ( n , p )$. It is required to test $\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }$ against $\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }$ at a significance level as close to $1 \%$ as possible, using a sample of size $n = 8,9$ or 10 . Use tables to find which value of $n$ gives such a test, stating the critical region for the test and the corresponding significance level.\\[0pt]
[4]\\
\hfill \mbox{\textit{OCR S2 2013 Q1 [4]}}