| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Calculate probability of Type I error |
| Difficulty | Standard +0.8 This S4 question requires understanding of hypothesis testing, significance level, and power functions. Part (a) is routine calculation, but part (b) requires algebraic manipulation of binomial probabilities to derive a specific form, which demands careful work. Parts (c) and (d) are straightforward applications. The derivation in (b) and the conceptual understanding of power functions elevate this above average difficulty, though it remains a standard S4 topic without requiring novel insight. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion |
A train company claims that the probability $p$ of one of its trains arriving late is 10\%. A regular traveller sets up the hypothesis $H_0: p = 0.1$ and decides 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 decides to reject $H_0$ if $x \ge 2$.
\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 Q3 [9]}}