Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (p = 1/6 vs p > 1/6), standard significance level, and given data (n=15, x=5). Students need to calculate P(X ≥ 5) and compare to 0.10, which is routine application of the binomial test procedure with no conceptual complications or multi-step reasoning required. Slightly easier than average due to clear setup and small n allowing direct calculation.
2 Arvind uses an ordinary fair 6-sided die to play a game. He believes he has a system to predict the score when the die is thrown. Before each throw of the die, he writes down what he thinks the score will be. He claims that he can write the correct score more often than he would if he were just guessing. His friend Laxmi tests his claim by asking him to write down the score before each of 15 throws of the die. Arvind writes the correct score on exactly 5 out of 15 throws.
Test Arvind's claim at the \(10 \%\) significance level.
Valid comparison; For valid comparison with 0.9 (0.9102 > 0.9 seen the previous M1 and A1 can be recovered)
[Reject \(H_0\)] There is evidence (at the 10% level) that Arvind can predict scores
FTA1
Not definite, e.g. not 'He can predict' or 'Claim true'; In context and no contradictions
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: P(\text{correct}) = \frac{1}{6}$ and $H_1: P(\text{correct}) > \frac{1}{6}$ | B1 | Allow $p = \frac{1}{6}$; Allow $p > \frac{1}{6}$ |
| $1 - \binom{15}{4}\left(\frac{5}{6}\right)^{11}\left(\frac{1}{6}\right)^4 + \binom{15}{3}\left(\frac{5}{6}\right)^{12}\left(\frac{1}{6}\right)^3 + \binom{15}{2}\left(\frac{5}{6}\right)^{13}\left(\frac{1}{6}\right)^2 + 15\left(\frac{5}{6}\right)^{14}\cdot\frac{1}{6} + \left(\frac{5}{6}\right)^{15}$ | M1 | Expression must be seen; Allow one end error |
| $0.0898$ or $0.0897$ (3 sf) | A1 | SC if M0 scored allow SCB1 for 0.0898 or 0.0897 |
| $0.0898 < 0.1$ | M1 | Valid comparison; For valid comparison with 0.9 (0.9102 > 0.9 seen the previous M1 and A1 can be recovered) |
| [Reject $H_0$] There is evidence (at the 10% level) that Arvind can predict scores | FTA1 | Not definite, e.g. not 'He can predict' or 'Claim true'; In context and no contradictions |
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2 Arvind uses an ordinary fair 6-sided die to play a game. He believes he has a system to predict the score when the die is thrown. Before each throw of the die, he writes down what he thinks the score will be. He claims that he can write the correct score more often than he would if he were just guessing. His friend Laxmi tests his claim by asking him to write down the score before each of 15 throws of the die. Arvind writes the correct score on exactly 5 out of 15 throws.
Test Arvind's claim at the $10 \%$ significance level.\\
\hfill \mbox{\textit{CAIE S2 2022 Q2 [5]}}