| Exam Board | OCR MEI |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2011 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Find minimum sample size for Type II error constraint |
| Difficulty | Challenging +1.2 This is a standard S4/Further Statistics question requiring application of Type I/II error concepts to find sample size and critical value. Part (i) is definitional recall, part (ii) involves routine manipulation of two normal distribution equations (using z-scores for the 2% and 95% conditions), and part (iii) requires sketching a step function. While it requires careful algebraic manipulation and understanding of power functions, it follows a well-established template for this topic with no novel insights needed. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Type I error: rejecting null hypothesis when it is true | B1, B1 | Allow B1 out of 2 for P(...) |
| Type II error: accepting null hypothesis when it is false | B1, B1 | Allow B1 out of 2 for P(...) |
| OC: P(accepting null hypothesis as a function of the parameter under investigation) | B1, B1 | P(Type II error \ |
| Power: P(rejecting null hypothesis as a function of the parameter under investigation) | B1, B1 | P(Type I error \ |
| [8] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(X \sim N(\mu, 25)\), \(H_0: \mu=94\), \(H_1: \mu > 94\) | ||
| Require \(0.02 = P(\text{reject } H_0 \mid \mu=94) = P(\bar{X} > c \mid \mu=94)\) | M1 | |
| \(= P\left(N(94, 25/n) > c\right) = P\left(N(0,1) > \frac{c-94}{5/\sqrt{n}}\right)\) | M1, M1 | for first expression; for standardising |
| \(\therefore \frac{c-94}{5/\sqrt{n}} = 2.054\) | B1 | for 2.054 |
| Also require \(0.95 = P(\text{reject } H_0 \mid \mu=97)\) | ||
| \(= P\left(N(97, 25/n) > c\right) = P\left(N(0,1) > \frac{c-97}{5/\sqrt{n}}\right)\) | M1, M1 | for first expression; for standardising |
| \(\therefore \frac{c-97}{5/\sqrt{n}} = -1.645\) | B1 | for \(-1.645\) |
| \(\therefore c = 94 + \frac{10.27}{\sqrt{n}}\) and \(c = 97 - \frac{8.225}{\sqrt{n}}\) | M1, A1 | two equations; both correct (FT any previous errors) |
| Attempt to solve; \(c = 95.666\) [allow 95.7 or awrt] | M1, A1 c.a.o. | |
| \(\sqrt{n} = 6.165\), \(n = 38.01\); take \(n\) as "next integer up" from candidate's value | A1 c.a.o., A1 | |
| [13] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Power function: step function from 0, with step marked at 94, to height marked as 1 | G1, G1, G1 | Zero out of 3 if step is wrong way round |
| [3] |
# Question 3 (4769 June 2011):
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Type I error: rejecting null hypothesis when it is true | B1, B1 | Allow B1 out of 2 for P(...) |
| Type II error: accepting null hypothesis when it is false | B1, B1 | Allow B1 out of 2 for P(...) |
| OC: P(accepting null hypothesis as a function of the parameter under investigation) | B1, B1 | P(Type II error \| true value of parameter) scores B1+B1 |
| Power: P(rejecting null hypothesis as a function of the parameter under investigation) | B1, B1 | P(Type I error \| true value of parameter) scores B1+B1; "1−OC" as definition scores zero |
| **[8]** | | |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $X \sim N(\mu, 25)$, $H_0: \mu=94$, $H_1: \mu > 94$ | | |
| Require $0.02 = P(\text{reject } H_0 \mid \mu=94) = P(\bar{X} > c \mid \mu=94)$ | M1 | |
| $= P\left(N(94, 25/n) > c\right) = P\left(N(0,1) > \frac{c-94}{5/\sqrt{n}}\right)$ | M1, M1 | for first expression; for standardising |
| $\therefore \frac{c-94}{5/\sqrt{n}} = 2.054$ | B1 | for 2.054 |
| Also require $0.95 = P(\text{reject } H_0 \mid \mu=97)$ | | |
| $= P\left(N(97, 25/n) > c\right) = P\left(N(0,1) > \frac{c-97}{5/\sqrt{n}}\right)$ | M1, M1 | for first expression; for standardising |
| $\therefore \frac{c-97}{5/\sqrt{n}} = -1.645$ | B1 | for $-1.645$ |
| $\therefore c = 94 + \frac{10.27}{\sqrt{n}}$ and $c = 97 - \frac{8.225}{\sqrt{n}}$ | M1, A1 | two equations; both correct (FT any previous errors) |
| Attempt to solve; $c = 95.666$ [allow 95.7 or awrt] | M1, A1 c.a.o. | |
| $\sqrt{n} = 6.165$, $n = 38.01$; take $n$ as "next integer up" from candidate's value | A1 c.a.o., A1 | |
| **[13]** | | |
## Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Power function: step function from 0, with step marked at 94, to height marked as 1 | G1, G1, G1 | Zero out of 3 if step is wrong way round |
| **[3]** | | |
---3 (i) Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.\\
(ii) A market research organisation is designing a sample survey to investigate whether expenditure on everyday food items has increased in 2011 compared with 2010. For one of the populations being studied, the random variable $X$ is used to model weekly expenditure, in $\pounds$, on these items in 2011, where $X$ is Normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$. As the corresponding mean value in 2010 was 94 , the hypotheses to be examined are
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 94 \\
& \mathrm { H } _ { 1 } : \mu > 94
\end{aligned}$$
By comparison with the corresponding 2010 value, $\sigma ^ { 2 }$ is assumed to be 25 .\\
The following criteria for the survey are laid down.
\begin{itemize}
\item If in fact $\mu = 94$, the probability of concluding that $\mu > 94$ must be only $2 \%$
\item If in fact $\mu = 97$, the probability of concluding that $\mu > 94$ must be $95 \%$
\end{itemize}
A random sample of size $n$ is to be taken and the usual Normal test based on $\bar { X }$ is to be used, with a critical value of $c$ such that $\mathrm { H } _ { 0 }$ is rejected if the value of $\bar { X }$ exceeds $c$. Find $c$ and the smallest value of $n$ that is required.\\
(iii) Sketch the power function of an ideal test for examining the hypotheses in part (ii).
\hfill \mbox{\textit{OCR MEI S4 2011 Q3 [24]}}