| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2004 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a straightforward application of standard S4 procedures: calculating confidence intervals for mean (using t-distribution) and variance (using chi-squared), then interpreting results. The calculations are routine with clear data, though it requires knowledge of two different confidence interval formulas and careful interpretation in part (b). |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks |
|---|---|
| (a) \(\bar{x} = 123.1\) | B1 |
| \(s = 5.87745\ldots\) | B1 |
| (NB: \(\sum x = 1231; \sum x^2 = 151847\)) | |
| (i) 95% confidence interval is given by \(123.1 \pm 2.262 \times \frac{5.87745\ldots}{\sqrt{10}}\) | M1 |
| 2.262 | B1 |
| i.e: \((118.8958\ldots, 127.30418\ldots)\) | A1 ft |
| AWRT (119, 127) | A1 A1 |
| (ii) 95% confidence interval is given by \(\frac{9 \times 5.87745\ldots^2}{19.023} < \sigma^2 < \frac{9 \times 5.87745\ldots^2}{2.700}\) use of \(\frac{(n-1)s^2}{\sigma^2}\) | M1 |
| 19.023 | B1 |
| 2.700 | B1 |
| i.e; \((16.34336\ldots, 115.14806\ldots)\) | A1ft |
| AWRT (16.3, 115) | A1 A1 |
| (13) | |
| (b) 130 is just outside confidence interval | B1 |
| 16 is just outside confidence interval | B1 |
| Thus supervisor should be concerned about the speed of the new typist | B1 |
| (3) | |
| (16 marks) |
| **(a)** $\bar{x} = 123.1$ | B1 |
| $s = 5.87745\ldots$ | B1 |
| (NB: $\sum x = 1231; \sum x^2 = 151847$) | |
| **(i)** 95% confidence interval is given by $123.1 \pm 2.262 \times \frac{5.87745\ldots}{\sqrt{10}}$ | M1 |
| 2.262 | B1 |
| i.e: $(118.8958\ldots, 127.30418\ldots)$ | A1 ft |
| AWRT (119, 127) | A1 A1 |
| **(ii)** 95% confidence interval is given by $\frac{9 \times 5.87745\ldots^2}{19.023} < \sigma^2 < \frac{9 \times 5.87745\ldots^2}{2.700}$ use of $\frac{(n-1)s^2}{\sigma^2}$ | M1 |
| 19.023 | B1 |
| 2.700 | B1 |
| i.e; $(16.34336\ldots, 115.14806\ldots)$ | A1ft |
| AWRT (16.3, 115) | A1 A1 |
| | (13) |
| **(b)** 130 is just outside confidence interval | B1 |
| 16 is just outside confidence interval | B1 |
| Thus supervisor should be concerned about the speed of the new typist | B1 |
| | (3) |
| | (16 marks) |6. A supervisor wishes to cheek the typing speed of a new typist. On 10 randomly selected occasions, the supervisor records the time taken for the new typist to type 100 words. The results, in seconds, are given below.
$$110,125,130,126,128,127,118,120,122,125$$
The supervisor assumes that the time taken to type 100 words is normally distributed.
\begin{enumerate}[label=(\alph*)]
\item Calculate a 95\% confidence interval for
\begin{enumerate}[label=(\roman*)]
\item the mean,
\item the variance\\
of the population of times taken by this typist to type 100 words.
The supervisor requires the average time needed to type 100 words to be no more than 130 seconds and the standard deviation to be no more than 4 seconds.
\end{enumerate}\item Comment on whether or not the supervisor should be concerned about the speed of the new typist.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2004 Q6 [16]}}