| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Standard +0.8 This is a comprehensive S4 hypothesis testing question requiring multiple calculations (Type I error, power function, critical region determination) across two different scenarios. While the individual techniques are standard for Further Maths S4, the question demands sustained accuracy across 7 parts, understanding of power vs significance level trade-offs, and interpretation of results. The critical region calculation in part (d) requires trial-and-error with cumulative binomial probabilities to get 'as close as possible to 0.05', which is more demanding than routine textbook exercises. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.3\) (or 0.7), \(H_1: p < 0.3\) (or \(> 0.7\)) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{Type I error}) = P(X < 6 \mid p = 0.30) = P(X \leq 5) = 0.0766\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Power \(= P(Y \leq 5 \mid Y \sim B(30, 0.2)) = 0.4275\) | M1 A1 | |
| (ii) Power \(= P(Y \leq 5 \mid Y \sim B(30, 0.1)) = 0.9268\) | A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| We require \(c\) such that \(P(C \leq c) \approx 0.05\) | M1 | |
| \(P(C \leq 10) = 0.0789\), \(P(C \leq 9) = 0.0402\) \(\therefore\) critical region is \(C \leq 9\) | A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Size \(= 0.0402\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Power \(= P(D \leq 9 \mid D \sim B(50, 0.2)) = 0.4437\) | B1 | |
| (ii) Power \(= P(D \leq 9 \mid D \sim B(50, 0.1)) = 0.9755\) | B1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Advantage: second test is more powerful | B1 | |
| Disadvantage: second test involves greater sample size, \(\therefore\) more expensive or takes longer | B1 | (2 marks) (13 marks total) |
# Question 3:
## Part (a)
$H_0: p = 0.3$ (or 0.7), $H_1: p < 0.3$ (or $> 0.7$) | B1 | (1 mark)
## Part (b)
Let $X$ = number who contract virus. Under $H_0$, $X \sim B(30, 0.3)$
$P(\text{Type I error}) = P(X < 6 \mid p = 0.30) = P(X \leq 5) = 0.0766$ | M1 A1 | (2 marks)
## Part (c)
(i) Power $= P(Y \leq 5 \mid Y \sim B(30, 0.2)) = 0.4275$ | M1 A1 |
(ii) Power $= P(Y \leq 5 \mid Y \sim B(30, 0.1)) = 0.9268$ | A1 | (3 marks)
## Part (d)
Let $C$ = number who contract virus. Under $H_0$, $C \sim B(50, 0.3)$
We require $c$ such that $P(C \leq c) \approx 0.05$ | M1 |
$P(C \leq 10) = 0.0789$, $P(C \leq 9) = 0.0402$ $\therefore$ critical region is $C \leq 9$ | A1 | (2 marks)
## Part (e)
Size $= 0.0402$ | B1 | (1 mark)
## Part (f)
(i) Power $= P(D \leq 9 \mid D \sim B(50, 0.2)) = 0.4437$ | B1 |
(ii) Power $= P(D \leq 9 \mid D \sim B(50, 0.1)) = 0.9755$ | B1 | (2 marks)
## Part (g)
Advantage: second test is more powerful | B1 |
Disadvantage: second test involves greater sample size, $\therefore$ more expensive or takes longer | B1 | (2 marks) **(13 marks total)**
---3. A certain vaccine is known to be only $70 \%$ effective against a particular virus; thus $30 \%$ of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.
\begin{enumerate}[label=(\alph*)]
\item Write down suitable hypotheses for this test.
\item Find the probability of making a Type I error.
\item Find the power of this test if the new vaccine is
\begin{enumerate}[label=(\roman*)]
\item $80 \%$ effective,
\item $90 \%$ effective.
An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than $70 \%$ effective. They required the probability of a Type I error to be as close as possible to 0.05 .
\end{enumerate}\item Find the critical region for this test.
\item State the size of this critical region.
\item Find the power of this test if the new vaccine is
\begin{enumerate}[label=(\roman*)]
\item $80 \%$ effective,
\item $90 \%$ effective.
\end{enumerate}\item Give one advantage and one disadvantage of the second test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q3 [13]}}