| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unbiased estimator from summary statistics |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas. Part (i) requires the unbiased variance formula s²=Σw²-(Σw)²/n/(n-1), which is direct substitution. Part (ii) applies CLT to state W̄~N(μ, σ²/70) using the sample mean and variance from part (i) - both are routine bookwork with minimal problem-solving required. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05d Sample mean as random variable |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\bar{w} = 100.8 \div 14 = 7.2\) | B1 | 7.2 seen or implied |
| \(\frac{938.70}{14} - \bar{w}^2 = [15.21]\) | M1 | Use \(\sum w^2\) – their \(\bar{w}^2\) |
| \(\times \frac{14}{13}\) | M1 | Multiply by \(n/(n-1)\) |
| \(= \frac{16.38}{...}\) | A1 | 4 |
| \((ST: 7.2, 16.38 ÷ 70)\) | B1 | Normal stated |
| (ii) | \([N(7.2, 0.234)]\) | |
| B1V | 3 | Variance [their \((1)\div + 70]\), allow arithmetic slip |
(i) $\bar{w} = 100.8 \div 14 = 7.2$ | B1 | 7.2 seen or implied
$\frac{938.70}{14} - \bar{w}^2 = [15.21]$ | M1 | Use $\sum w^2$ – their $\bar{w}^2$
$\times \frac{14}{13}$ | M1 | Multiply by $n/(n-1)$
$= \frac{16.38}{...}$ | A1 | 4 | Answer, a.r.t. 16.4
| | $(ST: 7.2, 16.38 ÷ 70)$ | B1 | Normal stated
(ii) | | $[N(7.2, 0.234)]$ | B1V | Mean their $\bar{w}$ √
| | | B1V | 3 | Variance [their $(1)\div + 70]$, allow arithmetic slip4 A set of observations of a random variable $W$ can be summarised as follows:
$$n = 14 , \quad \Sigma w = 100.8 , \quad \Sigma w ^ { 2 } = 938.70 .$$
(i) Calculate an unbiased estimate of the variance of $W$.\\
(ii) The mean of 70 observations of $W$ is denoted by $\bar { W }$. State the approximate distribution of $\bar { W }$, including unbiased estimate(s) of any parameter(s).
\hfill \mbox{\textit{OCR S2 2007 Q4 [7]}}