| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Estimate from summary statistics |
| Difficulty | Moderate -0.3 This is a straightforward application of standard formulas for unbiased estimates (sample mean and variance with n-1 denominator) followed by a routine normal probability calculation. It requires recall of formulas and careful arithmetic but no problem-solving insight, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\hat{\mu} = \bar{x} = 38\) | B1 | 38 stated separately |
| \(\frac{\Sigma x^2}{10} - 38^2 \quad [=16.2]\) | M1 | Use of \(\frac{\Sigma x^2}{n} - \bar{x}^2\). Correct single formula: M2 |
| \(\times \frac{10}{9}\) to get 18 | M1, A1 | Multiply by 10/9. 18 or a.r.t. 18.0 only. If single formula, divisor of 9 seen anywhere gets second M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Phi\!\left(\dfrac{40-38}{\sqrt{18}}\right) = \Phi(0.4714) = \mathbf{0.3187}\) | M1 | Standardise with their \(\mu\) and \(\sigma\); allow cc, \(\sqrt{\text{errors}}\) |
| A1 | Answer, a.r.t. 0.319. Allow a.r.t. 0.311 [0.3106] from 16.2. \(\sqrt{10}\) used: M0 |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\hat{\mu} = \bar{x} = 38$ | B1 | 38 stated separately |
| $\frac{\Sigma x^2}{10} - 38^2 \quad [=16.2]$ | M1 | Use of $\frac{\Sigma x^2}{n} - \bar{x}^2$. Correct single formula: M2 |
| $\times \frac{10}{9}$ to get **18** | M1, A1 | Multiply by 10/9. 18 or a.r.t. 18.0 only. If single formula, divisor of 9 seen anywhere gets second M1 |
**[4 marks]**
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Phi\!\left(\dfrac{40-38}{\sqrt{18}}\right) = \Phi(0.4714) = \mathbf{0.3187}$ | M1 | Standardise with their $\mu$ and $\sigma$; allow cc, $\sqrt{\text{errors}}$ |
| | A1 | Answer, a.r.t. 0.319. Allow a.r.t. 0.311 [0.3106] from 16.2. $\sqrt{10}$ used: M0 |
**[2 marks]**
---2 A random variable $C$ has the distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$. A random sample of 10 observations of $C$ is obtained, and the results are summarised as
$$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$
(i) Calculate unbiased estimates of $\mu$ and $\sigma ^ { 2 }$.\\
(ii) Hence calculate an estimate of the probability that $C > 40$.
\hfill \mbox{\textit{OCR S2 2013 Q2 [6]}}