| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (H₀: p=0.8 vs H₁: p>0.8). Students need to calculate P(X≥23) using tables or formula, compare to 10% significance level, and make a conclusion. Part (b) requires basic critical thinking about model assumptions. The question is slightly easier than average because it's a standard textbook procedure with generous significance level and the critical value calculation is routine. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: p = 0.8\), \(H_1: p > 0.8\) | B1 | |
| \(P(X \geq 23) = \binom{25}{23} \times 0.2^2 \times 0.8^{23} + \binom{25}{24} \times 0.2 \times 0.8^{24} + 0.8^{25} = 0.070835 + 0.0236118 + 0.0037779\) | M1 | No end errors. Expression must be seen or supported by enough figures to be convinced \(B(25, 0.8)\) used. Accept correct \(\Sigma\) notation. |
| \(= 0.0982\) | A1 | SC B1 for 0.0982 unsupported |
| \(0.0982 < 0.1\) | M1 | Valid comparison; their 0.0982 must be a tail probability |
| There is sufficient evidence to suggest that \(p\) has increased | ftA1 | No contradictions. In context, non-definite. Note: CR method includes \(P(X \geq 23)\) so M1 A1 as above, and \(P(X \geq 22) = 0.234 > 0.1\) with at least one probability comparison with 0.1 needed to find CR of 23,24,25 (so 23 in CR) M1 A1ft as above. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Not suitable as model does not allow for more than 25 employees to use the canteen / Not suitable as uses a sample instead of all employees / Not suitable doesn't include all employees / Not suitable as 30 is only just bigger than 25 should have used 30; OR Suitable as owner knows that not all employees use the canteen, or similar | B1 | Need both (i.e. suitable or not suitable plus reason) |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p = 0.8$, $H_1: p > 0.8$ | B1 | |
| $P(X \geq 23) = \binom{25}{23} \times 0.2^2 \times 0.8^{23} + \binom{25}{24} \times 0.2 \times 0.8^{24} + 0.8^{25} = 0.070835 + 0.0236118 + 0.0037779$ | M1 | No end errors. Expression must be seen or supported by enough figures to be convinced $B(25, 0.8)$ used. Accept correct $\Sigma$ notation. |
| $= 0.0982$ | A1 | SC B1 for 0.0982 unsupported |
| $0.0982 < 0.1$ | M1 | Valid comparison; their 0.0982 must be a tail probability |
| There is sufficient evidence to suggest that $p$ has increased | ftA1 | No contradictions. In context, non-definite. Note: CR method includes $P(X \geq 23)$ so M1 A1 as above, and $P(X \geq 22) = 0.234 > 0.1$ with at least one probability comparison with 0.1 needed to find CR of 23,24,25 (so 23 in CR) **M1 A1ft** as above. |
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Not suitable as model does not allow for more than 25 employees to use the canteen / Not suitable as uses a sample instead of all employees / Not suitable doesn't include all employees / Not suitable as 30 is only just bigger than 25 should have used 30; OR Suitable as owner knows that not all employees use the canteen, or similar | B1 | Need both (i.e. suitable or not suitable plus reason) |
---3 A factory owner models the number of employees who use the factory canteen on any day by the distribution $\mathrm { B } ( 25 , p )$. In the past the value of $p$ was 0.8 . A new menu is introduced in the canteen and the owner wants to test whether the value of $p$ has increased.
On a randomly chosen day he notes that the number of employees who use the canteen is 23 .
\begin{enumerate}[label=(\alph*)]
\item Use the binomial distribution to carry out the test at the $10 \%$ significance level.
\item Given that there are 30 employees at the factory comment on the suitability of the owner's model.\\
\includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-04_2713_33_111_2016}\\
\includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-05_2716_29_107_22}
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q3 [6]}}