| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Justifying CLT for confidence intervals |
| Difficulty | Standard +0.3 Part (a) is a standard confidence interval calculation using the t-distribution with given sample statistics. Part (b) tests understanding that the Central Limit Theorem justifies using normal approximation for the sample mean when n=50 is sufficiently large, regardless of the population distribution. This is routine application of CLT with straightforward conceptual explanation, slightly easier than average due to being a direct textbook application. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | 34.711 |
| Answer | Marks |
|---|---|
| = 34.711 ± 0.424 or (34.287, 35.135) | B1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3.4 | Allow 34.29 to 35.13 or 35.14 | |
| 1 | (b) | 50 is a sufficiently large sample to apply the CLT |
| Answer | Marks |
|---|---|
| the sample mean is approximately Normal | B1* |
| Answer | Marks |
|---|---|
| [2] | 2.2b |
| 2.4 | For mention of central limit theorem |
| For full statement (including CLT) | No credit if CLT not |
Question 1:
1 | (a) | 34.711
± 1.96
1.53
×
50
= 34.711 ± 0.424 or (34.287, 35.135) | B1
M1
M1
A1
[4] | 1.1
3.3
1.1
3.4 | Allow 34.29 to 35.13 or 35.14
1 | (b) | 50 is a sufficiently large sample to apply the CLT
which states that for large samples the distribution of
the sample mean is approximately Normal | B1*
*B1
[2] | 2.2b
2.4 | For mention of central limit theorem
For full statement (including CLT) | No credit if CLT not
mentioned1 When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .
\begin{enumerate}[label=(\alph*)]
\item Determine a 95\% confidence interval for the population mean head circumference of newborn female babies.
\item Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2021 Q1 [6]}}