| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Find minimum sample size |
| Difficulty | Standard +0.3 This is a straightforward S3 confidence interval question requiring standard formulas: part (a) uses basic sample mean and variance calculations (unbiased estimator with n-1), while part (b) applies the standard sample size formula n = (z*σ/E)² with given values. Both parts are direct applications of textbook methods with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Marks | Guidance |
| (a) Let \(x\) represent slider time. \(\sum w = 1433\) \(\bar{x} = \frac{1433}{5} = 286.7\) \(\sum x^2 = 442375\) \(S^2 = \frac{1}{4}\left\{442375 - \frac{1433^2}{5}\right\} = 7662.5\) | B1, M1, A1 (4) | |
| (b) \(P( | μ - \bar{x} | < 2σ = 0.95\) \(\therefore \frac{2σ}{\sqrt{n}} = 1.96\) \(\therefore n = 1.96^2 = 1.96 \times 100^2 = 96.04\) Sample size (\(>\))97 required |
| Content | Marks | Guidance |
|---------|-------|----------|
| **(a)** Let $x$ represent slider time. $\sum w = 1433$ $\bar{x} = \frac{1433}{5} = 286.7$ $\sum x^2 = 442375$ $S^2 = \frac{1}{4}\left\{442375 - \frac{1433^2}{5}\right\} = 7662.5$ | B1, M1, A1 (4) | |
| **(b)** $P(|μ - \bar{x}| < 2σ = 0.95$ $\therefore \frac{2σ}{\sqrt{n}} = 1.96$ $\therefore n = 1.96^2 = 1.96 \times 100^2 = 96.04$ **Sample size** ($>$)97 required | M1, B1, A1, M1, A1 (6) | Solving for n |
---A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are
205 \quad 310 \quad 405 \quad 195 \quad 320.
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. [4]
\end{enumerate}
It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the minimum sample size required. [6]
\end{enumerate}
(Total 10 marks)
\hfill \mbox{\textit{Edexcel S3 2005 Q6 [10]}}