Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (p=1/3), small n=12 allowing direct calculation from tables, and standard 5% significance level. The setup is explicit and requires only routine application of the binomial test procedure, making it slightly easier than average but still requiring proper statistical reasoning.
3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the \(5 \%\) level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.
Allow \(\pi\), but \(\mu = \frac{1}{3}\) etc B1. Any other letter, B0. Not \(\mu = 4\)
\(H_1: p \neq \frac{1}{3}\) [or 0.33 or better]
One-tailed, or no symbol, B1 max. *(if in doubt, consult)*
\(\alpha\): \(B(12, \frac{1}{3})\) stated or implied
M1
\(B(12, \frac{1}{3})\) stated or implied, allow for \(N(4,8/3)\), \(Po(4)\). *If N used, or \(P(\leq 7)\) or \(P(= 7)\), no more marks*
\(P(\geq 7) = 1 - 0.9336 = 0.0664\)
A1
Probability in range \([0.066, 0.067]\)
\(> 0.025\)
A1
Explicit comparison with 0.025, or \(2p\) with 0.05. 1-tailed: A0 here regardless of value
\(\beta\): CR is \(\geq 8\), 7 not in CR
A1
Needs explicit comparison of 7 with CV. Need to be clear that CR is being used
Probability is 0.0188
A1
Must be \(\geq 7\), 0.019 or 0.0188 or better, allow 0.9812
Do not reject \(H_0\). Insufficient evidence that statement is false.
M1
Needs correct method, including like-with-like, correct tail, \(\geq 7\) (or \(\leq 6\)). If CV, needs right tail. Allow from 1-tail: 0.9812 or 0.0188 or 0.0476
\(A1\sqrt{}\)
A1 needs "evidence" or equivalent. \(\sqrt{}\) on their \(p\)/CR. Withhold if answer refers only to \(p\).
[7]
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p = \frac{1}{3}$ [or 0.33 or better] | B2 | Allow $\pi$, but $\mu = \frac{1}{3}$ etc B1. Any other letter, B0. Not $\mu = 4$ |
| $H_1: p \neq \frac{1}{3}$ [or 0.33 or better] | | One-tailed, or no symbol, B1 max. *(if in doubt, consult)* |
| $\alpha$: $B(12, \frac{1}{3})$ stated or implied | M1 | $B(12, \frac{1}{3})$ stated or implied, allow for $N(4,8/3)$, $Po(4)$. *If N used, or $P(\leq 7)$ or $P(= 7)$, no more marks* |
| $P(\geq 7) = 1 - 0.9336 = 0.0664$ | A1 | Probability in range $[0.066, 0.067]$ |
| $> 0.025$ | A1 | Explicit comparison with 0.025, or $2p$ with 0.05. 1-tailed: A0 here regardless of value |
| $\beta$: CR is $\geq 8$, 7 not in CR | A1 | Needs explicit comparison of 7 with CV. Need to be clear that CR is being used |
| Probability is 0.0188 | A1 | Must be $\geq 7$, 0.019 or 0.0188 or better, allow 0.9812 |
| Do not reject $H_0$. Insufficient evidence that statement is false. | M1 | Needs correct method, including like-with-like, correct tail, $\geq 7$ (or $\leq 6$). If CV, needs right tail. Allow from 1-tail: 0.9812 or 0.0188 or 0.0476 |
| | $A1\sqrt{}$ | A1 needs "evidence" or equivalent. $\sqrt{}$ on their $p$/CR. Withhold if answer refers only to $p$. |
| | **[7]** | |
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3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the $5 \%$ level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.
\hfill \mbox{\textit{OCR S2 2012 Q3 [7]}}