| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Pooled variance estimation |
| Difficulty | Standard +0.3 This is a straightforward application of standard formulas for unbiased estimates and pooled statistics. Part (a) requires direct substitution into memorized formulas for sample mean and variance. Part (b) involves combining two samples using the pooled mean and variance formulas, which is a standard S3 technique but requires careful bookkeeping across multiple steps. The question is slightly easier than average because it's purely procedural with no conceptual challenges or problem-solving required. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a) | M1 A1, M1 A1 | \(\mu = \bar{x} = \frac{1039}{30} = 34.6\); \(\hat{\sigma}^2 = s^2 = \frac{30}{29}\left(\frac{65393}{30} - 34.63^2\right) = 1014.1\) |
| Part (b) | M1 A1, M1 A1, M1 A1, M1 A1, (10) | \(\frac{\sum x}{20} = 32.0\) \(\therefore \sum x = 640\) \(\mu\) for combined sample \(= \frac{1039 + 640}{50} = 33.6\); \(963.4 = \frac{30}{19}\left(\frac{\sum x^2}{20} - 32.0^2\right)\) giving \(\sum x^2 = 38784.6\); \(\hat{\sigma}^2\) for combined sample \(= \frac{50}{49}\left(\frac{65393 + 38784.6}{50} - 33.58^2\right) = 975.4\) |
**Part (a)** | M1 A1, M1 A1 | $\mu = \bar{x} = \frac{1039}{30} = 34.6$; $\hat{\sigma}^2 = s^2 = \frac{30}{29}\left(\frac{65393}{30} - 34.63^2\right) = 1014.1$
**Part (b)** | M1 A1, M1 A1, M1 A1, M1 A1, (10) | $\frac{\sum x}{20} = 32.0$ $\therefore \sum x = 640$ $\mu$ for combined sample $= \frac{1039 + 640}{50} = 33.6$; $963.4 = \frac{30}{19}\left(\frac{\sum x^2}{20} - 32.0^2\right)$ giving $\sum x^2 = 38784.6$; $\hat{\sigma}^2$ for combined sample $= \frac{50}{49}\left(\frac{65393 + 38784.6}{50} - 33.58^2\right) = 975.4$A student collected data on the number of text messages, $t$, sent by 30 students in her year group in the previous week. Her results are summarised as follows:
$\Sigma t = 1039$, $\Sigma t^2 = 65393$.
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the mean and variance of the number of text messages sent by these students per week. [4]
\end{enumerate}
Another student collected similar data for 20 different students and calculated unbiased estimates of the mean and variance of 32.0 and 963.4 respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Calculate unbiased estimates of the mean and variance for the combined sample of 50 students. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q4 [10]}}