| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2003 |
| Session | June |
| Marks | 8 |
| 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 +0.3 This is a straightforward application of hypothesis testing with known standard deviation. Part (i) requires calculating P(X̄ < 1.7 | μ = 2.1) using the sampling distribution, and part (ii) requires P(X̄ ≥ 1.7 | μ = 1.5). Both involve standard normal distribution calculations with clear parameters given. The question is slightly easier than average because all values are provided explicitly, requiring only correct application of formulas rather than problem-solving or conceptual insight. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(X < 1.7) = \Phi\left(\frac{1.7 - 2.1}{0.9/\sqrt{20}}\right)\) | \(B1\) | For identifying prob Type I error |
| \(M1\) | For standardising | |
| \(= 1 - \Phi(1.9876)\) | \(A1\) | For correct standardising and correct area |
| \(= 0.0234\) | \(A1\) | For correct final answer |
| (ii) \(P(\text{Type II error}) = P(X > 1.7)\) | \(B1\) | For identifying prob for Type II error |
| \(= 1 - \Phi\left(\frac{1.7 - 1.5}{0.9/\sqrt{20}}\right)\) | \(M1\) | For standardising using 1.5 and their 1.7 |
| \(= 1 - \Phi(0.9938) = 0.160\) | \(A1, A1\) | For correct standardising and correct area |
| For correct final answer |
**(i)** $P(X < 1.7) = \Phi\left(\frac{1.7 - 2.1}{0.9/\sqrt{20}}\right)$ | $B1$ | For identifying prob Type I error
| $M1$ | For standardising
$= 1 - \Phi(1.9876)$ | $A1$ | For correct standardising and correct area
$= 0.0234$ | $A1$ | For correct final answer
**(ii)** $P(\text{Type II error}) = P(X > 1.7)$ | $B1$ | For identifying prob for Type II error
$= 1 - \Phi\left(\frac{1.7 - 1.5}{0.9/\sqrt{20}}\right)$ | $M1$ | For standardising using 1.5 and their 1.7
$= 1 - \Phi(0.9938) = 0.160$ | $A1, A1$ | For correct standardising and correct area
| | For correct final answer5 Over a long period of time it is found that the time spent at cash withdrawal points follows a normal distribution with mean 2.1 minutes and standard deviation 0.9 minutes. A new system is tried out, to speed up the procedure. The null hypothesis is that the mean time spent is the same under the new system as previously. It is decided to reject the null hypothesis and accept that the new system is quicker if the mean withdrawal time from a random sample of 20 cash withdrawals is less than 1.7 minutes. Assume that, for the new system, the standard deviation is still 0.9 minutes, and the time spent still follows a normal distribution.\\
(i) Calculate the probability of a Type I error.\\
(ii) If the mean withdrawal time under the new system is actually 1.5 minutes, calculate the probability of a Type II error.
\hfill \mbox{\textit{CAIE S2 2003 Q5 [8]}}