| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | State meaning of Type I error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question covering standard S2 content: calculating sample statistics from a frequency table, performing a one-tailed z-test (or t-test), and explaining Type I error with a diagram. All parts are routine applications of learned procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02g Calculate mean and standard deviation2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Number of cars caught speeding | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Number of days | 5 | 7 | 8 | 10 | 5 | 2 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| variance \(= \frac{1}{39}\left(1815 - \frac{261^2}{40}\right) = 2.87\) | B1, M1, A1 (3 marks) | Correct mean. Substituting in formula from tables. Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| reduction in cars exceeding speed limit | B1, M1, A1, B1, B1ft (5 marks) | Correct \(H_1\). Standardising attempt must have \(\sqrt{40}\). Correct test statistic. Correct CV or finding area on LHS of their \(z\). Correct conclusion, reject \(H_0\) not enough, must compare \(+\) with \(+\) or \(-\) with \(-\) |
| Answer | Marks |
|---|---|
| shaded area on LHS labelled as \(5\%\) and Type I error. | B1, B1, B1 (3 marks) |
**Part (i):**
mean $= 6.53$ $(6.525)$
variance $= \frac{1}{39}\left(1815 - \frac{261^2}{40}\right) = 2.87$ | B1, M1, A1 (3 marks) | Correct mean. Substituting in formula from tables. Correct answer
**Part (ii):**
$H_1: \mu < 7.2$
test statistic $z = \frac{6.525-7.2}{\sqrt{\frac{2.871}{40}}} = -2.52$
critical value $z = \pm 1.645$ or $\pm 1.64$ or $\pm 1.65$
reduction in cars exceeding speed limit | B1, M1, A1, B1, B1ft (5 marks) | Correct $H_1$. Standardising attempt must have $\sqrt{40}$. Correct test statistic. Correct CV or finding area on LHS of their $z$. Correct conclusion, reject $H_0$ not enough, must compare $+$ with $+$ or $-$ with $-$
**Part (iii):**
saying there is a reduction in the number of speeding cars, when there isn't
normal curve with mean $7.2$ shown and small shaded area on LHS
shaded area on LHS labelled as $5\%$ and Type I error. | B1, B1, B1 (3 marks)7 The number of cars caught speeding on a certain length of motorway is 7.2 per day, on average. Speed cameras are introduced and the results shown in the following table are those from a random selection of 40 days after this.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Number of cars caught speeding & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of days & 5 & 7 & 8 & 10 & 5 & 2 & 3 \\
\hline
\end{tabular}
\end{center}
(i) Calculate unbiased estimates of the population mean and variance of the number of cars per day caught speeding after the speed cameras were introduced.\\
(ii) Taking the null hypothesis $\mathrm { H } _ { 0 }$ to be $\mu = 7.2$, test at the $5 \%$ level whether there is evidence that the introduction of speed cameras has resulted in a reduction in the number of cars caught speeding.\\
(iii) State what is meant by a Type I error in words relating to the context of the test in part (ii). Without further calculation, illustrate on a suitable diagram the region representing the probability of this Type I error.
\hfill \mbox{\textit{CAIE S2 2006 Q7 [11]}}