| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Simultaneous critical region and Type II error |
| Difficulty | Standard +0.8 This question requires understanding of hypothesis testing with normal distributions, calculating critical values using the sampling distribution of the mean, and interpreting Type II error probability geometrically. Part (ii) particularly demands insight into the relationship between power, Type II error, and the alternative hypothesis mean, going beyond routine test execution. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(-1.645 = \frac{c - 3.2}{1.4/\sqrt{10}}\) | M1 | For standardising, must have sq rt. and z value |
| B1 | For \(\pm 1.645\) used | |
| \(c = 2.47\) | A1 | For 2.47 |
| Rejection region is \(\bar{x} < 2.47\) | A1 ft | 4 |
| (ii) \(m < 2.47\) | B1 ft | 1 |
**(i)** $-1.645 = \frac{c - 3.2}{1.4/\sqrt{10}}$ | M1 | For standardising, must have sq rt. and z value
| B1 | For $\pm 1.645$ used
$c = 2.47$ | A1 | For 2.47
Rejection region is $\bar{x} < 2.47$ | A1 ft | 4 | For inequality correct way round (ft their 2.47 but must be <3.2)
**(ii)** $m < 2.47$ | B1 ft | 1 | ft on their (i)
---2 In summer the growth rate of grass in a lawn has a normal distribution with mean 3.2 cm per week and standard deviation 1.4 cm per week. A new type of grass is introduced which the manufacturer claims has a slower growth rate. A hypothesis test of this claim at the $5 \%$ significance level was carried out using a random sample of 10 lawns that had the new grass. It may be assumed that the growth rate of the new grass has a normal distribution with standard deviation 1.4 cm per week.\\
(i) Find the rejection region for the test.\\
(ii) The probability of making a Type II error when the actual value of the mean growth rate of the new grass is $m \mathrm {~cm}$ per week is less than 0.5 . Use your answer to part (i) to write down an inequality for $m$.
\hfill \mbox{\textit{CAIE S2 2007 Q2 [5]}}