| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2003 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find power function or power value |
| Difficulty | Challenging +1.2 This is a standard S4 hypothesis testing question requiring knowledge of binomial distributions, test size, and power functions. Part (b) involves algebraic manipulation of binomial probabilities but follows a predictable pattern. The calculations are routine for this module, though the power function derivation requires careful algebra. This is typical examination material for Further Maths S4 rather than requiring novel insight. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - 0.8891 = 0.1109\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - (P(0) + P(1) + P(2))\) | M1 | |
| \(= 1 - ((1-p)^{10} + 12p(1-p)^{11} + 66p^2(1-p)^{10})\) | M1A1 | |
| \(= 1 - (1-p)^{10}((1-p)^2 + 12p(1-p) + 66p^2)\) | cso | A1 |
| \(= 1 - (1-p)^{10}(1 + 10p + 55p^2)\) | given | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - 0.5583 = 0.442\) | M1A1 | |
| \(1 - 0.00281 = 0.997\) | A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| The test is more discriminating for the larger value of \(p\) | B1 | (1) |
## Part (a)
$1 - 0.8891 = 0.1109$ | B1 | (1)
## Part (b)
$1 - (P(0) + P(1) + P(2))$ | M1 |
$= 1 - ((1-p)^{10} + 12p(1-p)^{11} + 66p^2(1-p)^{10})$ | M1A1 |
$= 1 - (1-p)^{10}((1-p)^2 + 12p(1-p) + 66p^2)$ | cso | A1 |
$= 1 - (1-p)^{10}(1 + 10p + 55p^2)$ | given | (4)
## Part (c)(i)
$1 - 0.5583 = 0.442$ | M1A1 |
$1 - 0.00281 = 0.997$ | A1 | (3)
## Part (c)(ii)
The test is more discriminating for the larger value of $p$ | B1 | (1)
---A train company claims that the probability $p$ of one of its trains arriving late is 10\%. A regular traveller on the company's trains believes that the probability is greater than 10\% and decides to test this by randomly selecting 12 trains and recording the number $X$ of trains that were late. The traveller sets up the hypotheses H$_0$: $p = 0.1$ and H$_1$: $p > 0.1$ and accepts the null hypothesis if $x \leq 2$.
\begin{enumerate}[label=(\alph*)]
\item Find the size of the test. [1]
\item Show that the power function of the test is
$$1 - (1 - p)^{10}(1 + 10p + 55p^2).$$ [4]
\item Calculate the power of the test when
\begin{enumerate}[label=(\roman*)]
\item $p = 0.2$,
\item $p = 0.6$. [3]
\end{enumerate}
\item Comment on your results from part (c). [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2003 Q3 [9]}}