| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Moderate -0.3 This is a straightforward two-tail z-test application with given values that plug directly into the standard formula. Part (i) requires calculating a test statistic and comparing to critical values (routine procedure), while part (ii) tests basic understanding that significance level equals Type I error probability (definitional recall). No problem-solving insight or complex reasoning required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \mu = 12.5\); \(H_1: \mu \neq 12.5\) | B1 | |
| \(\frac{13.5 - 12.5}{4.2 \div \sqrt{50}}\) | M1 | allow \(4.2 \div 50\) |
| \(= 1.68(4)\) | A1 | |
| \(`1.684` < 1.96\) | M1 | comp 1.96 allow comp 1.645 if \(H_1: \mu > 12.5\); or comp \(1 -\) ('1.684') with 0.025 |
| No evidence that mean time has changed | A1ft [5] | No contradictions; ft their 1.684, but not comp 1.645 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| 0.05 | B1 [1] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu = 12.5$; $H_1: \mu \neq 12.5$ | B1 | |
| $\frac{13.5 - 12.5}{4.2 \div \sqrt{50}}$ | M1 | allow $4.2 \div 50$ |
| $= 1.68(4)$ | A1 | |
| $`1.684` < 1.96$ | M1 | comp 1.96 allow comp 1.645 if $H_1: \mu > 12.5$; or comp $1 -$ ('1.684') with 0.025 |
| No evidence that mean time has changed | A1ft [5] | No contradictions; ft their 1.684, but not comp 1.645 |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| 0.05 | B1 [1] | |
---4 In the past, the time spent by customers in a certain shop had mean 12.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 13.5 minutes.\\
(i) Assuming that the standard deviation remains at 4.2 minutes, test at the $5 \%$ significance level whether the mean time spent by customers in the shop has changed.\\
(ii) Another random sample of 50 customers is chosen and a similar test at the $5 \%$ significance level is carried out. State the probability of a Type I error.
\hfill \mbox{\textit{CAIE S2 2016 Q4 [6]}}