| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Calculate probability of Type I error |
| Difficulty | Moderate -0.3 This is a straightforward hypothesis testing question covering standard S4 definitions and calculations. Parts (a)-(b) are pure recall, part (c) is a routine binomial probability calculation, part (d) requires algebraic manipulation but follows a standard method, and parts (e)-(g) are direct applications of formulas. While it requires understanding of Type I/II errors and power functions, all techniques are standard bookwork with no novel problem-solving required. Slightly easier than average due to the high proportion of definition/recall marks. |
| 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 |
Define
\begin{enumerate}[label=(\alph*)]
\item a Type I error, [1]
\item the size of a test. [1]
\end{enumerate}
Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times $X$ that Jane is recorded.
The probability of Jane stating the correct symbol is denoted by $p$.
To test the hypothesis H₀: $p = 0.25$ against H₁: $p > 0.25$, a critical region of $X > 6$ is used.
\begin{enumerate}[label=(\alph*)]
\item[(c)] Find the size of this test. [3]
\item Show that the power function of this test is $8p^7 - 7p^8$. [3]
\end{enumerate}
Given that $p = 0.3$, calculate
\begin{enumerate}[label=(\alph*)]
\item[(e)] the power of this test, [1]
\item the probability of a Type II error. [2]
\item Suggest two ways in which you might reduce the probability of a Type II error. [2]
\end{enumerate}
(Total 12 marks)
\hfill \mbox{\textit{Edexcel S4 Q5 [13]}}