| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 9 |
| 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 S2 hypothesis testing question requiring routine application of binomial distribution tables to find critical regions and significance levels. Part (d) adds a minor twist by changing to a one-tailed test, but all parts follow textbook procedures with no novel problem-solving required. Slightly easier than average due to straightforward application of learned techniques. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(H_0: p = \frac{1}{2}\) and \(H_1: p \ne \frac{1}{2}\) | B1 | |
| (b) let X = no. with mobile phones \(\therefore X \sim B(25, \frac{1}{4})\); \(P(X \le 7) = 0.0216; P(X \le 17) = 0.9784\); \(\therefore\) C.R. is \(X \le 7\) or \(X \ge 18\) | M1 M1 A1 A1 | |
| (c) \(0.0216 + 0.0216 = 0.0432\) | A1 | |
| (d) \(H_0: p = \frac{1}{2}\) and \(H_1: p < \frac{1}{2}\); \(P(X \le 8) = 0.0539\); more than 5% \(\therefore\) not significant | B1 M1 A1 | (9 marks total) |
(a) $H_0: p = \frac{1}{2}$ and $H_1: p \ne \frac{1}{2}$ | B1 |
(b) let X = no. with mobile phones $\therefore X \sim B(25, \frac{1}{4})$; $P(X \le 7) = 0.0216; P(X \le 17) = 0.9784$; $\therefore$ C.R. is $X \le 7$ or $X \ge 18$ | M1 M1 A1 A1 |
(c) $0.0216 + 0.0216 = 0.0432$ | A1 |
(d) $H_0: p = \frac{1}{2}$ and $H_1: p < \frac{1}{2}$; $P(X \le 8) = 0.0539$; more than 5% $\therefore$ not significant | B1 M1 A1 | (9 marks total)
---3. A primary school teacher finds that exactly half of his year 6 class have mobile phones.
He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.
\begin{enumerate}[label=(\alph*)]
\item State the hypotheses that he should use.
\item Find the largest critical region for this test such that the probability in each "tail" is less than $2.5 \%$.
\item Determine the significance level of this test.
He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
\item Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than $5 \%$.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [9]}}