| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed test critical region |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring standard binomial critical region calculations at a given significance level, followed by routine Type I and Type II error probability computations. While it involves multiple parts, each step follows textbook procedures without requiring novel insight or complex reasoning. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X \leqslant n)\) \((n \leqslant 20)\) attempted, using \(B(20,\, 0.95)\) | M1 | OE |
| \(P(X \leqslant 17)\) or \(P(X \leqslant 16)\) attempted, using \(B(20,\, 0.95)\) | M1 | OE |
| \((P(X \leqslant 17)) = 0.0755\) and \((P(X \leqslant 16)) = 0.0159\) | A1 | OE (0.925 and 0.984) both correct |
| Rejection region is \(X \leqslant 16\) or \(X < 17\) | A1 | Dependent on M1M1 and previous answers correct to at least 0.075/0.076 and 0.016 or 0.92/0.93 and 0.98. Correct unsupported answers of 0.0755 and 0.0159 OE scores M1M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.0159\) | B1 | FT *their* rejection region, from Binomial in \(a\), if $P(X \text{ in rejection region}) < 0.025 |
| Answer | Marks | Guidance |
|---|---|---|
| Use of \(B(20, 0.7)\) | M1 | |
| \(P(X > 16 \mid p = 0.7)\) | M1 | Correct method using \(B(20, 0.7)\) |
| \(= 0.107\) | A1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X \leqslant n)$ $(n \leqslant 20)$ attempted, using $B(20,\, 0.95)$ | M1 | OE |
| $P(X \leqslant 17)$ or $P(X \leqslant 16)$ attempted, using $B(20,\, 0.95)$ | M1 | OE |
| $(P(X \leqslant 17)) = 0.0755$ and $(P(X \leqslant 16)) = 0.0159$ | A1 | OE (0.925 and 0.984) both correct |
| Rejection region is $X \leqslant 16$ or $X < 17$ | A1 | Dependent on M1M1 and previous answers correct to at least 0.075/0.076 and 0.016 **or** 0.92/0.93 and 0.98. Correct unsupported answers of 0.0755 and 0.0159 OE scores M1M1A0 |
## Question 7(b):
$0.0159$ | B1 | FT *their* rejection region, from Binomial in $a$, if $P(X \text{ in rejection region}) < 0.025
**Total: 1 mark**
---
## Question 7(c):
Use of $B(20, 0.7)$ | M1 |
$P(X > 16 \mid p = 0.7)$ | M1 | Correct method using $B(20, 0.7)$
$= 0.107$ | A1 |
**Total: 3 marks**7 A national survey shows that $95 \%$ of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who use social media is less than the national percentage. He chooses a random sample of 20 students at his college and notes the number who use social media. He then carries out a test at the $2 \%$ significance level.
\begin{enumerate}[label=(\alph*)]
\item Find the rejection region for the test.
\item Find the probability of a Type I error.
\item Jimmy believes that the true percentage at Arvin's college is $70 \%$. Assuming that Jimmy is correct, find the probability of a Type II error.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q7 [8]}}