| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | State meaning of Type II error |
| Difficulty | Standard +0.3 This is a straightforward one-sample z-test with large sample size (n=180), requiring standard hypothesis test procedure and a contextual definition of Type II error. The calculations are routine (z = 0.3/(2.8/√180) ≈ 1.44), and Type II error definition is recall-based. Slightly above average difficulty due to being a two-part question requiring both procedure and conceptual understanding, but no novel insight needed. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(H_0: \mu = 3\) | B1 |
| \(H_1: \mu > 3\) | ||
| Test statistic \(z = \frac{3.3-3}{2.8/\sqrt{179}} = 1.43\) | M1 | Standardising attempt with sq rt in denom. Correct z value accept rounding to 1.44 from \(\sqrt{180}\) (OR alt method finding crit value 3.344 M1 A1) |
| A1 | ||
| Critical value \(z = 1.645\) | M1 | Comparing with \(z = 1.645\) (or z consistent with their \(H_1\)) or eqiv comparison of areas |
| Not enough evidence to support the claim | A1ft [5] | Correct answer, ft their z. No contradictions. (OR compare C.V 3.344 with 3.3 M1 A1ft) |
| (ii) | Say no extra weight loss when there is. | B2 [2] |
(i) | $H_0: \mu = 3$ | B1 | Both hypotheses correct
| $H_1: \mu > 3$ |
Test statistic $z = \frac{3.3-3}{2.8/\sqrt{179}} = 1.43$ | M1 | Standardising attempt with sq rt in denom. Correct z value accept rounding to 1.44 from $\sqrt{180}$ (OR alt method finding crit value 3.344 M1 A1)
| | A1 |
Critical value $z = 1.645$ | M1 | Comparing with $z = 1.645$ (or z consistent with their $H_1$) or eqiv comparison of areas
Not enough evidence to support the claim | A1ft [5] | Correct answer, ft their z. No contradictions. (OR compare C.V 3.344 with 3.3 M1 A1ft)
(ii) | Say no extra weight loss when there is. | B2 [2] | Correct statement in context. SR₁ B1 for partially correct statement in context. SR₂ B1 for any true statement about a Type II error including accept H₀ when it is false
---4 People who diet can expect to lose an average of 3 kg in a month. In a book, the authors claim that people who follow a new diet will lose an average of more than 3 kg in a month. The weight losses of the 180 people in a random sample who had followed the new diet for a month were noted. The mean was 3.3 kg and the standard deviation was 2.8 kg .\\
(i) Test the authors' claim at the $5 \%$ significance level, stating your null and alternative hypotheses.\\
(ii) State what is meant by a Type II error in words relating to the context of the test in part (i).
\hfill \mbox{\textit{CAIE S2 2008 Q4 [7]}}