| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Find critical region for test |
| Difficulty | Standard +0.3 This is a straightforward one-tailed hypothesis test with known population standard deviation. Part (a) requires standard application of the normal distribution to find a critical region (routine calculation with z = 1.645), and part (b) involves converting total height to mean, computing a test statistic, and stating a conclusion. While it requires careful unit conversion and multiple steps, it follows a standard textbook procedure with no novel problem-solving or conceptual challenges beyond typical S3 material. |
| Spec | 2.05e Hypothesis test for normal mean: known variance |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(H_0: \mu = 5'6''\) \(H_1: \mu > 5'6''\) | B1 | |
| 5% level ∴ C.R. is \(z > 1.6449\) | B1 | |
| require = \(\frac{\overline{x}-66}{\frac{s}{\sqrt{150}}} > 1.6449\) | M2, A1 | |
| giving C.R. \(\overline{X} > 66.31\) inches | A1 | |
| (b) \(\overline{X} = \frac{832+32}{150} = 66.56\) | M1, A1 | |
| 66.56 > 66.31 ∴ reject \(H_0\) there is evidence that mean height of women is > 5'6'' | M1, A1 | (10) |
**(a)** $H_0: \mu = 5'6''$ $H_1: \mu > 5'6''$ | B1 |
5% level ∴ C.R. is $z > 1.6449$ | B1 |
require = $\frac{\overline{x}-66}{\frac{s}{\sqrt{150}}} > 1.6449$ | M2, A1 |
giving C.R. $\overline{X} > 66.31$ inches | A1 |
**(b)** $\overline{X} = \frac{832+32}{150} = 66.56$ | M1, A1 |
66.56 > 66.31 ∴ reject $H_0$ there is evidence that mean height of women is > 5'6'' | M1, A1 | (10) |
---3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches.
Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the $5 \%$ level of significance.
The total height of the females in the sample is 832 ft .
\item Carry out the test making your conclusion clear.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q3 [10]}}