| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Calculate probability of Type I error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring standard procedures: finding a critical region using normal distribution tables (part a), calculating Type I error probability from a given critical region (part b), and computing Type II error for a specific alternative hypothesis (part c). All parts follow textbook methods with no novel insight required, though it does test understanding of error types. Slightly easier than average due to the routine nature of the calculations. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(H_0 : \mu = 330\) \(H_1 : \mu < 330\) | B1 | 2.5 |
| \([\bar{X} \sim ]\text{N}\left(330, \left(\frac{8}{\sqrt{25}}\right)^2\right)\) | M1 | 3.3 |
| \(\text{P}(\bar{X} < C) = 0.05 \Rightarrow \frac{C - 330}{8/\sqrt{25}} = -1.6449\) | M1 | 3.4 |
| So \(C = 327.368\ldots\) and critical region is: \(\bar{X} <\) awrt 327 | A1 | 1.1b |
| (b) \(\bar{Y} \sim \text{N}\left(330, \left(\frac{8}{\sqrt{55}}\right)^2\right)\) and require \(2 \times \text{P}(\bar{Y} < 328)\) (o.e.) | M1 | 3.3 |
| \(= 0.063732\ldots\) awrt \(\mathbf{0.0637}\) | A1 | 1.1b |
| (c) \(\text{P}(\bar{X} > ''327.368...''\ | \) \(\mu = 325)\) or \(1 - \text{P}(\bar{X} < ''327.368...''\) \(\mu = 325)\) \(= 0.0694233\ldots\) awrt \(\mathbf{0.0694}\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2nd M1 for a correct equation for \(C\) Allow any \(z\) value that satisfies \(1.6 < | z | < 1.7\). If standardisation equation not seen, this mark may be implied by CV \(= \) awrt 327 or CR: \(<\)awrt 327 |
**(a)** $H_0 : \mu = 330$ $H_1 : \mu < 330$ | B1 | 2.5 |
$[\bar{X} \sim ]\text{N}\left(330, \left(\frac{8}{\sqrt{25}}\right)^2\right)$ | M1 | 3.3 |
$\text{P}(\bar{X} < C) = 0.05 \Rightarrow \frac{C - 330}{8/\sqrt{25}} = -1.6449$ | M1 | 3.4 |
So $C = 327.368\ldots$ and critical region is: $\bar{X} <$ awrt **327** | A1 | 1.1b | (4)
**(b)** $\bar{Y} \sim \text{N}\left(330, \left(\frac{8}{\sqrt{55}}\right)^2\right)$ and require $2 \times \text{P}(\bar{Y} < 328)$ (o.e.) | M1 | 3.3 |
$= 0.063732\ldots$ awrt $\mathbf{0.0637}$ | A1 | 1.1b | (2)
**(c)** $\text{P}(\bar{X} > ''327.368...''\|$ $\mu = 325)$ or $1 - \text{P}(\bar{X} < ''327.368...''$ $\mu = 325)$ $= 0.0694233\ldots$ awrt $\mathbf{0.0694}$ | M1, A1 | 3.4, 1.1b | (2)
**Notes:**
(a) B1 for both hypotheses in terms of $\mu$
1st M1 for stating or using the correct model – may be implied by use in later line. Condone $X$ or any letter for $\bar{X}$
2nd M1 for a correct equation for $C$ Allow any $z$ value that satisfies $1.6 < |z| < 1.7$. If standardisation equation not seen, this mark may be implied by CV $= $ awrt 327 or CR: $<$awrt 327
A1 for a correct CR allow just "$<$ awrt 327"" Condone e.g. $X < 327$ rather than $\bar{X} < 327$
Condone $\leq$:
(b) M1 for sight of correct model and attempt at P$(\bar{Y} < 328)$ (o.e.) Condone missing $2 \times$
A1 for awrt 0.0637 (correct answer scores 2 out of 2)
(c) M1 for a correct (ft) statement may be implied by sight of e.g. $Z > ''327.368...''-325 = 1.48..$
For $\mu = 325$ allow $\bar{X} \sim \text{N}(325, \ldots)$
Allow ft from a 2-tailed test in part (a)
A1 for awrt 0.0694 (correct answer scores 2 out of 2)
SC Sight of P$(328 < \bar{X} < 332 \mid \mu = 325)$ or $1 - \text{P}(\bar{X} < 328 \cup \bar{X} > 332 \mid \mu = 325)$ scores M1A0
**(Total: 8 marks)**
---\begin{enumerate}
\item A machine fills cartons with juice.
\end{enumerate}
The amount of juice in a carton is normally distributed with mean $\mu \mathrm { ml }$ and standard deviation 8 ml .
A manager wants to test whether or not the amount of juice in the cartons, $X \mathrm { ml }$, is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice $\bar { x } \mathrm { ml }$.\\
(a) Using a $5 \%$ level of significance, find the critical region of $\bar { X }$ for this test.
State your hypotheses clearly.
The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice $\bar { y } \mathrm { ml }$ and uses a test with a critical region of
$$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
(b) Find P (Type I error) for the Director's test.
When $\mu = 325 \mathrm { ml }$\\
(c) find P (Type II error) for the test in part (a)
\hfill \mbox{\textit{Edexcel FS1 2023 Q5 [8]}}