Challenging +1.2 This question requires applying the Central Limit Theorem to a binomial distribution, then performing a normal approximation with continuity correction. While it involves multiple conceptual steps (recognizing CLT applies, finding mean/variance of sample mean, standardizing), the execution is relatively straightforward once the approach is identified. It's moderately harder than average due to the CLT application rather than being a direct normal distribution problem.
1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.
\(P(\bar{X} > 1.4) = 1 - \Phi\left(\frac{1.4 - (+1/100) - 1.5}{\sqrt{1.275/50}}\right) = \Phi(0.5636) = 0.713\) or \(0.714\) or \(0.734\) without cc
B1, B1, M1, M1, A1
Mean and variance correct (OR if working with totals mean=75, var=63.75). Standardising with (correct) or without cc must have \(\sqrt{\text{var}/50}\) in denom (OR equiv standardisation using totals). Correct area i.e. \(> 0.5\). Correct answer, accept either.
$\text{mean} = 10 \times 0.15 = 1.5$, $\text{var} = 10 \times 0.15 \times 0.85 = 1.275$
$P(\bar{X} > 1.4) = 1 - \Phi\left(\frac{1.4 - (+1/100) - 1.5}{\sqrt{1.275/50}}\right) = \Phi(0.5636) = 0.713$ or $0.714$ or $0.734$ without cc | B1, B1, M1, M1, A1 | Mean and variance correct (OR if working with totals mean=75, var=63.75). Standardising with (correct) or without cc must have $\sqrt{\text{var}/50}$ in denom (OR equiv standardisation using totals). Correct area i.e. $> 0.5$. Correct answer, accept either.
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1 The random variable $X$ has the distribution $\mathrm { B } ( 10,0.15 )$. Find the probability that the mean of a random sample of 50 observations of $X$ is greater than 1.4.
\hfill \mbox{\textit{CAIE S2 2007 Q1 [5]}}