| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| 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 standard S4 hypothesis testing question requiring routine application of binomial critical regions and error probability calculations. While it involves multiple parts including Type I and Type II errors, all steps follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X =\) No of soft centres, \(X \sim B(20, 0.5)\) | ||
| Critical region \(X \leq 5\) or \(X \geq 15\) | B1 B1 | B1 for \(X \leq 5\); B1 for \(X \geq 15\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{Type I error}) = P(X \leq 5 \mid p=0.5) + P(X \geq 15 \mid p=0.5)\) | M1 | Adding their two CR together or a correct answer |
| \(= 0.0207 + 0.0207 = 0.0414\) | A1 | awrt 0.0414 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{Type II error}) = P(X < 15 \mid p=0.25) - P(X < 6 \mid p=0.25)\) | M1 | FT their CR |
| \(= 1 - 0.6172 = 0.3828\) | A1 | awrt 0.383 |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X =$ No of soft centres, $X \sim B(20, 0.5)$ | | |
| Critical region $X \leq 5$ or $X \geq 15$ | B1 B1 | B1 for $X \leq 5$; B1 for $X \geq 15$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{Type I error}) = P(X \leq 5 \mid p=0.5) + P(X \geq 15 \mid p=0.5)$ | M1 | Adding their two CR together or a correct answer |
| $= 0.0207 + 0.0207 = 0.0414$ | A1 | awrt 0.0414 |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{Type II error}) = P(X < 15 \mid p=0.25) - P(X < 6 \mid p=0.25)$ | M1 | FT their CR |
| $= 1 - 0.6172 = 0.3828$ | A1 | awrt 0.383 |
---3. A jar contains a large number of sweets which have either soft centres or hard centres. The jar is thought to contain equal proportions of sweets with soft centres and sweets with hard centres. A random sample of 20 sweets is taken from the jar and the number of sweets with hard centres is recorded.
\begin{enumerate}[label=(\alph*)]
\item Using a $5 \%$ level of significance, find the critical region for a two-tailed test of the hypothesis that there are equal proportions of sweets with soft centres and sweets with hard centres in the jar.
\item Calculate the probability of a Type I error for this test.
Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,
\item calculate the probability of a Type II error for this test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2016 Q3 [6]}}