| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Compare tests by Type II error probability |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring standard procedures: finding a critical region using normal distribution tables, calculating Type II error probability, and explaining conceptual relationships between sample size/significance level and error probabilities. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail2.05d Sample mean as random variable2.05e Hypothesis test for normal mean: known variance5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{c - 4000}{60/\sqrt{50}} = 1.645\) | M1, B1, A1 | Standardise unknown with \(\sqrt{50}\) or 50 [ignore RHS]. \(z = 1.645\) or \(−1.645\) seen. Wholly correct eqn, \(\checkmark\) on their \(z\) [\(1 - 1.645\): M1B1A0] |
| Solve: \(c = 4014\) [\(4013.958\)] | M1, A1, A1 | Solve to find \(c\). Value of \(c\), a.r.t. 4014. Answer "> 4014", allow ≥, \(\checkmark\) on their \(c\), needs M1M1 |
| (ii) Use "Type II is: accept when \(H_0\) false"; \(\beta = 1 - 4014\) | M1 dep, dep M1 | Standardise 4020 and 4014√, allow 60°, cc. With \(\sqrt{50}\) or 50 |
| \(\frac{4020 - 4014}{60/\sqrt{50}} = 0.7071\) [\(0.712\) from \(4013.958\)] | A1, M1 | Completely correct LHS, \(\checkmark\) on their \(c\). z-value in range [0.2375, 0.2405] |
| \(1 - \Phi(0.7071) = 0.240\) [\(0.238\) from \(4013.958\)] | A1 | Answer in range [0.2375, 0.2405] |
| (iii) Smaller | B1 | "Smaller" stated, no invalidating reason |
| Smaller cv, better test etc | B1 | Plausible reason |
| (iv) Smaller | B1 | "Smaller" stated, no invalidating reason |
| Smaller cv, larger prob of Type I etc | B1 | Plausible reason |
| (v) No, parent distribution known to be normal | B2 | "No" stated, convincing reason. SR: If B0, "No", reason that is not invalidating: B1 |
(i) $\frac{c - 4000}{60/\sqrt{50}} = 1.645$ | M1, B1, A1 | Standardise unknown with $\sqrt{50}$ or 50 [ignore RHS]. $z = 1.645$ or $−1.645$ seen. Wholly correct eqn, $\checkmark$ on their $z$ [$1 - 1.645$: M1B1A0]
Solve: $c = 4014$ [$4013.958$] | M1, A1, A1 | Solve to find $c$. Value of $c$, a.r.t. 4014. Answer "> 4014", allow ≥, $\checkmark$ on their $c$, needs M1M1
(ii) Use "Type II is: accept when $H_0$ false"; $\beta = 1 - 4014$ | M1 dep, dep M1 | Standardise 4020 and 4014√, allow 60°, cc. With $\sqrt{50}$ or 50
$\frac{4020 - 4014}{60/\sqrt{50}} = 0.7071$ [$0.712$ from $4013.958$] | A1, M1 | Completely correct LHS, $\checkmark$ on their $c$. z-value in range [0.2375, 0.2405]
$1 - \Phi(0.7071) = 0.240$ [$0.238$ from $4013.958$] | A1 | Answer in range [0.2375, 0.2405]
(iii) Smaller | B1 | "Smaller" stated, no invalidating reason
Smaller cv, better test etc | B1 | Plausible reason
(iv) Smaller | B1 | "Smaller" stated, no invalidating reason
Smaller cv, larger prob of Type I etc | B1 | Plausible reason
(v) No, parent distribution known to be normal | B2 | "No" stated, convincing reason. SR: If B0, "No", reason that is not invalidating: B1
---7 Three independent researchers, $A , B$ and $C$, carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, $X$ watts, is a normally distributed random variable with mean $\mu$ and standard deviation 60. Each researcher tests the null hypothesis $\mathrm { H } _ { 0 } : \mu = 4000$ against the alternative hypothesis $\mathrm { H } _ { 1 } : \mu > 4000$.
Researcher $A$ uses a sample of size 50 and a significance level of $5 \%$.\\
(i) Find the critical region for this test, giving your answer correct to 4 significant figures.
In fact the value of $\mu$ is 4020 .\\
(ii) Calculate the probability that Researcher $A$ makes a Type II error.\\
(iii) Researcher $B$ uses a sample bigger than 50 and a significance level of $5 \%$. Explain whether the probability that Researcher $B$ makes a Type II error is less than, equal to, or greater than your answer to part (ii).\\
(iv) Researcher $C$ uses a sample of size 50 and a significance level bigger than $5 \%$. Explain whether the probability that Researcher $C$ makes a Type II error is less than, equal to, or greater than your answer to part (ii).\\
(v) State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.
\hfill \mbox{\textit{OCR S2 2006 Q7 [18]}}