| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (lower tail) |
| Difficulty | Standard +0.3 This is a straightforward application of standard hypothesis testing procedures. Part (i) requires rearranging the z-statistic formula to find n (basic algebra), and part (ii) involves comparing z to a critical value from tables. Both parts are routine calculations with no conceptual challenges beyond standard S2 content. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{73.1 - 75.2}{\frac{5.7}{\sqrt{n}}} = -1.563\) | M1 | For standardising (with \(\sqrt{n}\)) |
| \(n = \{-1.563 \times 5.7 \div (-2.1)\}^2\) | A1 | Any correct expression for \(n\) or \(\sqrt{n}\). May be implied by ans. |
| \(n = 18\) | A1 | |
| Assume s.d. for the region is \(5.7\) | B1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0\): pop mean (or \(\mu\)) \(= 75.2\) | B1 | Both (could be stated in (i)) |
| \(H_1\): pop mean (or \(\mu\)) \(< 75.2\) | M1 | For comparison of \(z\) values / areas / \(x\) values |
| \(1.563\) comp \(1.555\) | A1 [3] | CWO. No contradictions |
| Evidence that plants shorter |
## Question 3:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{73.1 - 75.2}{\frac{5.7}{\sqrt{n}}} = -1.563$ | M1 | For standardising (with $\sqrt{n}$) |
| $n = \{-1.563 \times 5.7 \div (-2.1)\}^2$ | A1 | Any correct expression for $n$ or $\sqrt{n}$. May be implied by ans. |
| $n = 18$ | A1 | |
| Assume s.d. for the region is $5.7$ | B1 [4] | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0$: pop mean (or $\mu$) $= 75.2$ | B1 | Both (could be stated in (i)) |
| $H_1$: pop mean (or $\mu$) $< 75.2$ | M1 | For comparison of $z$ values / areas / $x$ values |
| $1.563$ comp $1.555$ | A1 [3] | CWO. No contradictions |
| Evidence that plants shorter | | |
---3 The heights of a certain variety of plant have been found to be normally distributed with mean 75.2 cm and standard deviation 5.7 cm . A biologist suspects that pollution in a certain region is causing the plants to be shorter than usual. He takes a random sample of $n$ plants of this variety from this region and finds that their mean height is 73.1 cm . He then carries out an appropriate hypothesis test.\\
(i) He finds that the value of the test statistic $z$ is - 1.563 , correct to 3 decimal places. Calculate the value of $n$. State an assumption necessary for your calculation.\\
(ii) Use this value of the test statistic to carry out the hypothesis test at the 6\% significance level.
\hfill \mbox{\textit{CAIE S2 2013 Q3 [7]}}