| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 13 |
| 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 standard hypothesis testing question requiring routine application of normal distribution theory and understanding of Type I/II errors. While it has multiple parts and requires careful calculation with the sampling distribution of the mean, all techniques are textbook procedures with no novel insight needed. Slightly above average difficulty due to the multi-part structure and Type II error calculations, but well within the scope of standard S2 material. |
| Spec | 2.05e Hypothesis test for normal mean: known variance5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(np > 5\) and \(nq > 5\); \(0.75n > 5\) is relevant; \(n ≥ 20\) | M2 | Use either \(nq > 5\) or \(npq > 5\); or "n = 20" seen: M1] [SR: If M0, use \(np\) > 5, "n = 20" only; or "n ≥ 20 only" |
| A1 | Final answer \(n ≥ 20\) or \(n = 20\) only | 3 |
| (b) (i) \(70.5 - \mu = 1.75\sigma\); \(\mu - 46.5 = 2.25\sigma\); Solve simultaneously; \(\mu = 60\); \(\sigma = ±154.5\) | M1 | Standardise once, and equate to Φ⁻¹, ± cc |
| A1 | Standardise twice, signs correct, cc correct | |
| B1 | Both 1.75 and 2.25 | |
| M1 | Correct solution method to get one variable | |
| A1 ∀ | μ, a.r.t. 60.0 or ± 154.5 | |
| A1 ∀ | σ, a.r.t. 6.00 [Wrong cc (below): A1 both] [SR: σ⁻¹: M1A0B1M1A1A0] | 6 |
| (ii) \(np = 60, npq = 36\) | M1 dep | \(np = 60\) and \(npq = 6^2\) or 6 |
| dep M1 | Solve to get \(q\) or \(p\) or \(n\) | |
| A1 ∀ | \(p = 0.4\) ∀ on wrong cc or ≥ | |
| A1 ∀ | \(n = 150\) ∀ on wrong cc or ≥ | 4 |
(a) $np > 5$ and $nq > 5$; $0.75n > 5$ is relevant; $n ≥ 20$ | M2 | Use either $nq > 5$ or $npq > 5$; or "n = 20" seen: M1] [SR: If M0, use $np$ > 5, "n = 20" only; or "n ≥ 20 only"
| A1 | Final answer $n ≥ 20$ or $n = 20$ only | 3
(b) (i) $70.5 - \mu = 1.75\sigma$; $\mu - 46.5 = 2.25\sigma$; Solve simultaneously; $\mu = 60$; $\sigma = ±154.5$ | M1 | Standardise once, and equate to Φ⁻¹, ± cc
| A1 | Standardise twice, signs correct, cc correct
| B1 | Both 1.75 and 2.25
| M1 | Correct solution method to get one variable
| A1 ∀ | μ, a.r.t. 60.0 or ± 154.5
| A1 ∀ | σ, a.r.t. 6.00 [Wrong cc (below): A1 both] [SR: σ⁻¹: M1A0B1M1A1A0] | 6
(ii) $np = 60, npq = 36$ | M1 dep | $np = 60$ and $npq = 6^2$ or 6
| dep M1 | Solve to get $q$ or $p$ or $n$
| A1 ∀ | $p = 0.4$ ∀ on wrong cc or ≥
| A1 ∀ | $n = 150$ ∀ on wrong cc or ≥ | 48 A random variable $Y$ is normally distributed with mean $\mu$ and variance 12.25. Two statisticians carry out significance tests of the hypotheses $\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0$.\\
(i) Statistician $A$ uses the mean $\bar { Y }$ of a sample of size 23, and the critical region for his test is $\bar { Y } > 64.20$. Find the significance level for $A$ 's test.\\
(ii) Statistician $B$ uses the mean of a sample of size 50 and a significance level of $5 \%$.
\begin{enumerate}[label=(\alph*)]
\item Find the critical region for $B$ 's test.
\item Given that $\mu = 65.0$, find the probability that $B$ 's test results in a Type II error.\\
(iii) Given that, when $\mu = 65.0$, the probability that $A$ 's test results in a Type II error is 0.1365 , state with a reason which test is better.
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2007 Q8 [13]}}