| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| 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 standard procedures: stating hypotheses, calculating a Type I error probability using normal distribution tables, and explaining error types. While it involves multiple parts, each step follows routine A-level statistics methodology with no novel problem-solving required. The calculations are direct applications of the Central Limit Theorem and z-score formulas, 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 |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): pop mean run time \(= 28.2\) mins; \(H_1\): pop mean run time \(< 28.2\) mins | B1 | Allow '\(\mu\)'. Not 'mean journey time' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{27 - 28.2}{4/\sqrt{40}} = -1.897\) | M1 | For standardising; Must have \(\sqrt{40}\) |
| \(\Phi(< -1.897) = 1 - \Phi(1.897)\) | M1 | For correct area consistent with these values |
| \(0.0289\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\) is not rejected so… | M1 | |
| Type II error can be made and Type I error cannot be made | A1 | Both needed (accept 'only a Type II error could be made') |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: pop mean run time $= 28.2$ mins; $H_1$: pop mean run time $< 28.2$ mins | B1 | Allow '$\mu$'. Not 'mean journey time' |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{27 - 28.2}{4/\sqrt{40}} = -1.897$ | M1 | For standardising; Must have $\sqrt{40}$ |
| $\Phi(< -1.897) = 1 - \Phi(1.897)$ | M1 | For correct area consistent with these values |
| $0.0289$ (3 sf) | A1 | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$ is not rejected so… | M1 | |
| Type II error can be made and Type I error cannot be made | A1 | Both needed (accept 'only a Type II error could be made') |7 In the past, the mean time for Jenny's morning run was 28.2 minutes. She does some extra training and she wishes to test whether her mean time has been reduced. After the training Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is less than 27 minutes she will conclude that the training has been effective. You may assume that, after the training, Jenny's run time has a standard deviation of 4.0 minutes.
\begin{enumerate}[label=(\alph*)]
\item State suitable null and alternative hypotheses for Jenny's test.
\item Find the probability that Jenny will make a Type I error.
\item Jenny found that the sample mean run time was 27.2 minutes.
Explain briefly whether it is possible for her to make a Type I error or a Type II error or both.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q7 [6]}}