| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | March |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (upper tail) |
| Difficulty | Moderate -0.8 This is a straightforward hypothesis testing question requiring only basic understanding of one-tail vs two-tail tests and comparison of a given z-value (2.41) to critical values. No calculations needed—the test statistic is provided, making this simpler than average A-level questions which typically require computing the test statistic from raw data. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One-tail because investigating whether "higher" | B1 | OE. Must have both parts |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): Population mean (or \(\mu\)) in city same as for others; \(H_1\): Population mean (or \(\mu\)) in city greater than for others | B1 FT | If (a) two-tail: \(H_0\): Pop mean (or \(\mu\)) in city same as for others; \(H_1\): Pop mean (or \(\mu\)) in region different from others |
| \(2.41 > 2.326\) or \(0.008 < 0.01\) or \(0.992 > 0.99\) | M1 | If (a) two-tail: \(2.41 < 2.576\) or \(0.992 < 0.995\) |
| There is evidence that buildings are higher [on average] | A1 FT | In context, not definite. No contradictions. If (a) two-tail: There is no evidence that the [average] height of buildings is different |
| 3 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| One-tail because investigating whether "higher" | B1 | OE. Must have both parts |
| | **1** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: Population mean (or $\mu$) in city same as for others; $H_1$: Population mean (or $\mu$) in city greater than for others | B1 FT | If **(a)** two-tail: $H_0$: Pop mean (or $\mu$) in city same as for others; $H_1$: Pop mean (or $\mu$) in region different from others |
| $2.41 > 2.326$ or $0.008 < 0.01$ or $0.992 > 0.99$ | M1 | If **(a)** two-tail: $2.41 < 2.576$ or $0.992 < 0.995$ |
| There is evidence that buildings are higher [on average] | A1 FT | In context, not definite. No contradictions. If **(a)** two-tail: There is no evidence that the [average] height of buildings is different |
| | **3** | |3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, $z$, and finds that $z = 2.41$.
\begin{enumerate}[label=(\alph*)]
\item Explain briefly whether he should use a one-tail test or a two-tail test.
\item Carry out the test at the $1 \%$ significance level.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q3 [4]}}