| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unbiased estimator from summary statistics |
| Difficulty | Moderate -0.8 Part (i) is direct application of standard formulas for unbiased estimators (sample mean and variance with n-1 denominator) requiring only substitution of given values. Part (ii) tests conceptual understanding of the Central Limit Theorem—recognizing that with n=100, the CLT applies regardless of X's distribution—but requires no calculation. This is straightforward bookwork with minimal problem-solving. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mu = 48.3\) | B1 | 48.3 seen |
| Biased estimate: 162.2016; can get B1M1M0 | M1 | |
| Multiply by \(n/(n-1)\) | M1 | |
| \(= 163.84\) | A1 | Answer, 164 or 163.8 or 163.84 |
| No, Central Limit theorem applies, so can assume distribution is normal | B2 | "No" with statement showing CLT is understood (though CLT does not need to be mentioned) [SR: No with reason that is not wrong: B1] |
$\mu = 48.3$ | B1 | 48.3 seen
Biased estimate: 162.2016; can get B1M1M0 | M1 |
Multiply by $n/(n-1)$ | M1 |
$= 163.84$ | A1 | Answer, 164 or 163.8 or 163.84
No, Central Limit theorem applies, so can assume distribution is normal | B2 | "No" with statement showing CLT is understood (though CLT does not need to be mentioned) [SR: No with reason that is not wrong: B1]1 A random sample of observations of a random variable $X$ is summarised by
$$n = 100 , \quad \Sigma x = 4830.0 , \quad \Sigma x ^ { 2 } = 249 \text { 509.16. }$$
(i) Obtain unbiased estimates of the mean and variance of $X$.\\
(ii) The sample mean of 100 observations of $X$ is denoted by $\bar { X }$. Explain whether you would need any further information about the distribution of $X$ in order to estimate $\mathrm { P } ( \bar { X } > 60 )$. [You should not attempt to carry out the calculation.]
\hfill \mbox{\textit{OCR S2 2007 Q1 [6]}}