| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 10 |
| 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 +0.3 This is a standard S2 hypothesis testing question requiring routine application of normal distribution theory to find a critical value, identify error types, and work backwards from power to find the true mean. All techniques are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| \(c = 74.63\) | Marks: M1 A1 A1 B1 A1 | Guidance: Equate standardised variable to \(\Phi^{-1}\), allow –, √12, 8 correct. 2.326 or a.r.t. 2.33 seen, signs must be correct. Answer, a.r.t. 74.6, cwo, allow ≤ or ≥ |
| (ii) (a) Answer: Type I error | Marks: B1 | Guidance: "Type I error" stated, needs evidence |
| (ii) (b) Answer: Correct | Marks: I | Guidance: "Correct" stated or clearly implied |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mu = 78.22\) | Marks: M1 *d ep M1 A1 | Guidance: \(\frac{c - \mu}{\text{SD}} = (\pm)\Phi^{-1}\), allow no √12 but not 80, not 8/√12. 0.8264. Correct including sign, √on their \(c\). Solve to find \(\mu\), dep, answer consistent with signs |
**(i)** **Answer:** $\frac{80 - c}{8/\sqrt{12}} = 2.326$
$c = 74.63$ | **Marks:** M1 A1 A1 B1 A1 | **Guidance:** Equate standardised variable to $\Phi^{-1}$, allow –, √12, 8 correct. 2.326 or a.r.t. 2.33 seen, signs must be correct. Answer, a.r.t. 74.6, cwo, allow ≤ or ≥
**(ii) (a)** **Answer:** Type I error | **Marks:** B1 | **Guidance:** "Type I error" stated, needs evidence
**(ii) (b)** **Answer:** Correct | **Marks:** I | **Guidance:** "Correct" stated or clearly implied
**(ii) (iii)** **Answer:** $\frac{74.63 - \mu}{8/\sqrt{12}} = -1.555$
Solve for $\mu$
$\mu = 78.22$ | **Marks:** M1 *d ep M1 A1 | **Guidance:** $\frac{c - \mu}{\text{SD}} = (\pm)\Phi^{-1}$, allow no √12 but not 80, not 8/√12. 0.8264. Correct including sign, √on their $c$. Solve to find $\mu$, dep, answer consistent with signs7 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 8 ^ { 2 } \right)$. The mean of a random sample of 12 observations of $X$ is denoted by $\bar { X }$. A test is carried out at the $1 \%$ significance level of the null hypothesis $\mathrm { H } _ { 0 } : \mu = 80$ against the alternative hypothesis $\mathrm { H } _ { 1 } : \mu < 80$. The test is summarised as follows: 'Reject $\mathrm { H } _ { 0 }$ if $\bar { X } < c$; otherwise do not reject $\mathrm { H } _ { 0 } { } ^ { \prime }$.\\
(i) Calculate the value of $c$.\\
(ii) Assuming that $\mu = 80$, state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
\begin{enumerate}[label=(\alph*)]
\item $\bar { X } = 74.0$,
\item $\bar { X } = 75.0$.\\
(iii) Independent repetitions of the above test, using the value of $c$ found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of $\mu$.
\end{enumerate}
\hfill \mbox{\textit{OCR S2 Q7 [10]}}