| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find or state significance level |
| Difficulty | Standard +0.8 This is a two-part question requiring (i) calculation of P(S≥8) under H₀ using binomial tables/calculator, and (ii) understanding Type II error across multiple scenarios with conditional probability. Part (ii) requires recognizing that Type II errors only occur when p=0.5 or 0.7, calculating β for each case, then using complement rule for 'at least one' across four tests—this involves sophisticated understanding of hypothesis testing concepts beyond routine application. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - P(\leq 7) = 0.0315\) or \(3.15\%\) | M1 | Clearly stated, or implied by any of 0.0083, 0.0933, 0.0103, 0.0576, 0.0744 |
| A1 | Ignore subsequent "therefore 5%" etc | |
| Total: 2 | Other answers 0/2 unless "\(1 - P(\leq 7)\)" explicitly seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{Type II error} \mid p = 0.5) = 0.6047\) | M1 | Explicit "\(P(\leq 7)\) for \(p = 0.5\)" [or 0.7] or 0.2120, 0.7805: M1 |
| \(P(\text{Type II error} \mid p = 0.7) = 0.0933\) | A1 | Both correct |
| \(\frac{1}{3}\times 0.6047 + \frac{1}{3}\times 0.0933\) | M1 | \(\frac{1}{3}\times\) one prob \(+ \frac{1}{3}\times\) other prob (*not* 0.9685) |
| \(P(\text{none of 4}) = 1-(1-0.2327)^4\) | M1 | \(1-(1-\text{ans})^4\) or equivalent binomial, not \(\frac{1}{3}\) |
| \(= \mathbf{0.653}\) | A1 | Allow in range \([0.653, 0.654]\) |
| Total: 5 | Independent. Independent [*not* \(\frac{16}{81}\) or \(\frac{65}{81}\)]. (all three M marks independent) |
## Question 8:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - P(\leq 7) = 0.0315$ or $3.15\%$ | M1 | Clearly stated, or implied by any of 0.0083, 0.0933, 0.0103, 0.0576, 0.0744 |
| | A1 | Ignore subsequent "therefore 5%" etc |
| **Total: 2** | | Other answers 0/2 unless "$1 - P(\leq 7)$" explicitly seen |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Type II error} \mid p = 0.5) = 0.6047$ | M1 | Explicit "$P(\leq 7)$ for $p = 0.5$" [or 0.7] or 0.2120, 0.7805: M1 |
| $P(\text{Type II error} \mid p = 0.7) = 0.0933$ | A1 | Both correct |
| $\frac{1}{3}\times 0.6047 + \frac{1}{3}\times 0.0933$ | M1 | $\frac{1}{3}\times$ one prob $+ \frac{1}{3}\times$ other prob (*not* 0.9685) |
| $P(\text{none of 4}) = 1-(1-0.2327)^4$ | M1 | $1-(1-\text{ans})^4$ or equivalent binomial, not $\frac{1}{3}$ |
| $= \mathbf{0.653}$ | A1 | Allow in range $[0.653, 0.654]$ |
| **Total: 5** | | Independent. Independent [*not* $\frac{16}{81}$ or $\frac{65}{81}$]. (all three M marks independent) |8 The random variable $S$ has the distribution $\mathrm { B } ( 14 , p )$. A significance test is carried out of the null hypothesis $\mathrm { H } _ { 0 } : p = 0.3$ against the alternative hypothesis $\mathrm { H } _ { 1 } : p > 0.3$. The critical region for the test is $S \geqslant 8$.\\
(i) Find the significance level of the test, correct to 3 significant figures.\\
(ii) It is given that, on each occasion that the test is carried out, the true value of $p$ is equally likely to be $0.3,0.5$ or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.
\hfill \mbox{\textit{OCR S2 2015 Q8 [7]}}