| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Hypothesis test for mean |
| Difficulty | Challenging +1.2 This is a straightforward application of the Central Limit Theorem to a binomial situation, requiring students to find the sampling distribution of the sample mean and determine critical regions using normal tables. While it involves multiple steps (finding mean/variance, applying CLT, finding critical values), each step follows a standard procedure taught in S3 with no novel insight required. It's moderately harder than average due to the multi-step nature and need to work with sampling distributions, but remains a textbook-style question. |
| Spec | 2.04d Normal approximation to binomial5.05a Sample mean distribution: central limit theorem |
8. A six-sided die is labelled with the numbers $1,2,3,4,5$ and 6
A group of 50 students want to test whether or not the die is fair for the number six.\\
The 50 students each roll the die 30 times and record the number of sixes they each obtain.\\
Given that $\bar { X }$ denotes the mean number of sixes obtained by the 50 students, and using
$$\mathrm { H } _ { 0 } : p = \frac { 1 } { 6 } \text { and } \mathrm { H } _ { 1 } : p \neq \frac { 1 } { 6 }$$
where $p$ is the probability of rolling a 6 ,
\begin{enumerate}[label=(\alph*)]
\item use the Central Limit Theorem to find an approximate distribution for $\bar { X }$, if $\mathrm { H } _ { 0 }$ is true.
\item Hence find, in terms of $\bar { X }$, the critical region for this test. Use a $5 \%$ level of significance.\\
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\hfill \mbox{\textit{Edexcel S3 2016 Q8 [7]}}