| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2020 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | State distribution of sample mean |
| Difficulty | Standard +0.8 This is a multi-part S3 question requiring CLT application to Poisson distributions. Part (a) is routine (applying CLT formula), but part (b) requires working backwards from confidence interval width to estimate λ, and part (c) involves understanding the probabilistic interpretation of confidence intervals—a conceptually subtle point that many students find challenging. The combination of computational work and conceptual understanding of confidence intervals elevates this above average difficulty. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05d Confidence intervals: using normal distribution |
6. The number of toasters sold by a shop each week may be modelled by a Poisson distribution with mean 4
A random sample of 35 weeks is taken and the mean number of toasters sold per week is found.
\begin{enumerate}[label=(\alph*)]
\item Write down the approximate distribution for the mean number of toasters sold per week from a random sample of 35 weeks.
The number of kettles sold by the shop each week may be modelled by a Poisson distribution with mean $\lambda$
A random sample of 40 weeks is taken and the mean number of kettles sold per week is found. The width of the $99 \%$ confidence interval for $\lambda$ is 2.6
\item Find an estimate for $\lambda$
A second, independent random sample of 40 weeks is taken and a second $99 \%$ confidence interval for $\lambda$ is found.
\item Find the probability that only one of these two confidence intervals contains $\lambda$\\
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\hfill \mbox{\textit{Edexcel S3 2020 Q6 [8]}}