| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Identify which error type was made |
| Difficulty | Standard +0.8 This is a multi-part hypothesis testing question requiring: (i) finding critical region using normal distribution with known variance, (ii) identifying Type I error, (iii) using binomial distribution to test meta-hypothesis about repeated tests, and (iv) calculating μ from Type II error probability (requiring reverse standardization). Part (iv) especially requires sophisticated understanding of power and working backwards from β=0.4, which goes beyond routine S2 exercises. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail2.05e Hypothesis test for normal mean: known variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(30 + 1.645 \times \frac{8}{\sqrt{18}} = 33.102\) | M1 | \(30 + z \times 8/\sqrt{18}\), allow \(\sqrt{}\) errors, cc |
| \(1.645\), requires \(+\) only | A1 | Allow \(\pm\) but not \(-\) only. No 18: 0 in this part. |
| \(33.1\) a.r.t. \(33.10\) | A1 | |
| so CR is \(\bar{X} > 33.1\) | A1\(\sqrt{}\) | \(\geq\) their RH CV\(\sqrt{}\); allow \(\leq\) their LH CV *as well*; \(>\), allow no letter or \(X\) but no other letter. Don't allow "accept if \(\leq 33.1\), reject if \(> 33.1\)". Inequality required in final line. |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Type I [error] | B1 | Nothing else unless it's just an amplification. Allow "Type 1" |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(B(20, 0.05)\): | M1 | \(B(20, 0.05)\) stated or implied. Not \(B(20, 1/5)\) |
| \(P(\geq 4) = 0.0159\) | A1 | Probability, a.r.t. \(0.016\) |
| so unlikely that \(\mu = 30\) | A1\(\sqrt{}\) | Justified conclusion, e.g. "I think \(\mu = 30\) as not less than \(0.01\)". FT on their \(p\). No reason: A0. Not over-assertive. But "I think \(\mu = 30\) as probability is small" is A0. |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{33.1 - \mu}{8/\sqrt{18}} = -0.253\) | M1 | Needs \(\Phi^{-1}\), their CV, SD right or same as in (i), allow cc. *Not* 30. Allow omission of \(\sqrt{18}\) only if omitted in (i). "\(1-\)" errors: can get M1A0A1 |
| Signs correct, can be implied by answer \(>\) their CV | A1 | |
| \(z\) in range \((\pm)[0.25, 0.26]\) | A1 | |
| \(\mu = 33.58\) | A1 | Final answer \(33.55 \leq \mu \leq 33.60\), 4 SF needed. Typically \(32.62\) probably gets 2/4. |
| [4] |
## Question 8:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $30 + 1.645 \times \frac{8}{\sqrt{18}} = 33.102$ | M1 | $30 + z \times 8/\sqrt{18}$, allow $\sqrt{}$ errors, cc |
| $1.645$, requires $+$ only | A1 | Allow $\pm$ but not $-$ only. No 18: 0 in this part. |
| $33.1$ a.r.t. $33.10$ | A1 | |
| so CR is $\bar{X} > 33.1$ | A1$\sqrt{}$ | $\geq$ their RH CV$\sqrt{}$; allow $\leq$ their LH CV *as well*; $>$, allow no letter or $X$ but no other letter. Don't allow "accept if $\leq 33.1$, reject if $> 33.1$". Inequality required in final line. |
| | **[4]** | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Type I [error] | B1 | Nothing else unless it's just an amplification. Allow "Type 1" |
| | **[1]** | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $B(20, 0.05)$: | M1 | $B(20, 0.05)$ stated or implied. Not $B(20, 1/5)$ |
| $P(\geq 4) = 0.0159$ | A1 | Probability, a.r.t. $0.016$ |
| so unlikely that $\mu = 30$ | A1$\sqrt{}$ | Justified conclusion, e.g. "I think $\mu = 30$ as not less than $0.01$". FT on their $p$. No reason: A0. Not over-assertive. But "I think $\mu = 30$ as probability is small" is A0. |
| | **[3]** | |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{33.1 - \mu}{8/\sqrt{18}} = -0.253$ | M1 | Needs $\Phi^{-1}$, their CV, SD right or same as in (i), allow cc. *Not* 30. Allow omission of $\sqrt{18}$ only if omitted in (i). "$1-$" errors: can get M1A0A1 |
| Signs correct, can be implied by answer $>$ their CV | A1 | |
| $z$ in range $(\pm)[0.25, 0.26]$ | A1 | |
| $\mu = 33.58$ | A1 | Final answer $33.55 \leq \mu \leq 33.60$, 4 SF needed. Typically $32.62$ probably gets 2/4. |
| | **[4]** | |8 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 8 ^ { 2 } \right)$. A test is carried out, at the $5 \%$ significance level, of $\mathrm { H } _ { 0 } : \mu = 30$ against $\mathrm { H } _ { 1 } : \mu > 30$, based on a random sample of size 18 .\\
(i) Find the critical region for the test.\\
(ii) If $\mu = 30$ and the outcome of the test is that $\mathrm { H } _ { 0 }$ is rejected, state the type of error that is made.
On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that $\mathrm { H } _ { 0 }$ is rejected. It is known that the value of $\mu$ remains the same throughout these 20 tests.\\
(iii) Find the probability that $\mathrm { H } _ { 0 }$ is rejected at least 4 times if $\mu = 30$. Hence state whether you think that $\mu = 30$, giving a reason.\\
(iv) Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of $\mu$, giving your answer correct to 4 significant figures.
\section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
\hfill \mbox{\textit{OCR S2 2012 Q8 [12]}}