| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | March |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Calculate probability of Type I error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring calculation of Type I error probability using normal distribution (z-test with known σ) and understanding the definition of Type II error. The concepts are standard S2 material with clear setup and routine calculations, though slightly above average difficulty due to requiring conceptual understanding of error types rather than pure computation. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): pop mean journey time \(= 35.2\) mins; \(H_1\): pop mean journey time \(< 35.2\) mins | B1 | Allow "\(\mu\)". Not "mean journey time" |
| \(\frac{34.7 - 35.2}{5.6/\sqrt{25}}\) \((= -0.446)\) | M1 | For standardising (\(\sqrt{25}\) needed) |
| \(\Phi(< \text{"}{-0.446}\text{"}) = 1 - \Phi(\text{"0.446"}) = 0.328\) (3 sf) | M1 A1 [4] | For correct area consistent with their working; as final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\) is rejected but Type II error can only be made if \(H_0\) is *not* rejected | B1 [1] | Allow just "\(H_0\) is rejected." oe |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: pop mean journey time $= 35.2$ mins; $H_1$: pop mean journey time $< 35.2$ mins | **B1** | Allow "$\mu$". Not "mean journey time" |
| $\frac{34.7 - 35.2}{5.6/\sqrt{25}}$ $(= -0.446)$ | **M1** | For standardising ($\sqrt{25}$ needed) |
| $\Phi(< \text{"}{-0.446}\text{"}) = 1 - \Phi(\text{"0.446"}) = 0.328$ (3 sf) | **M1 A1** [4] | For correct area consistent with their working; as final answer |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$ is rejected but Type II error can only be made if $H_0$ is *not* rejected | **B1** [1] | Allow just "$H_0$ is rejected." oe |
---3 In the past, Arvinder has found that the mean time for his journey to work is 35.2 minutes. He tries a different route to work, hoping that this will reduce his journey time. Arvinder decides to take a random sample of 25 journeys using the new route. If the sample mean is less than 34.7 minutes he will conclude that the new route is quicker. Assume that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.\\
(i) Find the probability that a Type I error occurs.\\
(ii) Arvinder finds that the sample mean is 34.5 minutes. Explain briefly why it is impossible for him to make a Type II error.
\hfill \mbox{\textit{CAIE S2 2016 Q3 [5]}}