| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a straightforward S4 hypothesis testing question covering standard definitions (size, power, Type II error) and routine binomial calculations. Parts (a)-(b) are pure recall, part (c) requires basic binomial probability tables, parts (d)-(e) apply standard concepts. While it's a multi-part question worth several marks, each component is textbook-standard with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) Size is the probability of \(H_0\) being rejected when it is in fact true. Or \(P(\text{reject } H_0 \mid H_0 \text{ is true})\) | B1 | oe |
| (b) The power of the test is the probability of rejecting \(H_0\) when \(H_1\) is true. Or \(P(\text{rejecting } H_0 \mid H_1 \text{ is true})\) / \(P(\text{rejecting } H_0 \mid H_0 \text{ is false})\) | B1 | oe |
| (c) \(X \sim B(12, 0.5)\) | B1 | |
| \(P(X \leq 2) = 0.0193\) | M1 | |
| \(P(X \geq 10) = 0.0193\) | ||
| \(\therefore\) critical region is \(\{X \leq 2 \cup X \geq 10\}\) | A1 A1 | |
| (d)(i) \(P(\text{Type II error}) = P(3 \leq X \leq 9 \mid p = 0.4)\) \(= P(X \leq 9) - P(X \leq 2)\) \(= 0.9972 - 0.0834 = 0.9138\) | M1, M1dep, A1 | 2nd M1 dependent on first M1; A1 cao |
| (d)(ii) Power \(= 1 - 0.9138 = 0.0862\) | B1ft | follow through from (i) |
| (e) Increase the sample size; Increase the significance level/larger critical region | B1, B1 |
# Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** Size is the probability of $H_0$ being rejected when it is in fact true. Or $P(\text{reject } H_0 \mid H_0 \text{ is true})$ | B1 | oe |
| **(b)** The power of the test is the probability of rejecting $H_0$ when $H_1$ is true. Or $P(\text{rejecting } H_0 \mid H_1 \text{ is true})$ / $P(\text{rejecting } H_0 \mid H_0 \text{ is false})$ | B1 | oe |
| **(c)** $X \sim B(12, 0.5)$ | B1 | |
| $P(X \leq 2) = 0.0193$ | M1 | |
| $P(X \geq 10) = 0.0193$ | | |
| $\therefore$ critical region is $\{X \leq 2 \cup X \geq 10\}$ | A1 A1 | |
| **(d)(i)** $P(\text{Type II error}) = P(3 \leq X \leq 9 \mid p = 0.4)$ $= P(X \leq 9) - P(X \leq 2)$ $= 0.9972 - 0.0834 = 0.9138$ | M1, M1dep, A1 | 2nd M1 dependent on first M1; A1 cao |
| **(d)(ii)** Power $= 1 - 0.9138 = 0.0862$ | B1ft | follow through from (i) |
| **(e)** Increase the sample size; Increase the significance level/larger critical region | B1, B1 | |
---\begin{enumerate}
\item Define, in terms of $\mathrm { H } _ { 0 }$ and/or $\mathrm { H } _ { 1 }$,\\
(a) the size of a hypothesis test,\\
(b) the power of a hypothesis test.
\end{enumerate}
The probability of getting a head when a coin is tossed is denoted by $p$.\\
This coin is tossed 12 times in order to test the hypotheses $\mathrm { H } _ { 0 } : p = 0.5$ against $\mathrm { H } _ { 1 } : p \neq 0.5$, using a 5\% level of significance.\\
(c) Find the largest critical region for this test, such that the probability in each tail is less than 2.5\%.\\
(d) Given that $p = 0.4$\\
(i) find the probability of a type II error when using this test,\\
(ii) find the power of this test.\\
(e) Suggest two ways in which the power of the test can be increased.\\
\hfill \mbox{\textit{Edexcel S4 2009 Q3 [12]}}