| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unbiased estimator from summary statistics |
| Difficulty | Standard +0.3 This is a straightforward S3 question requiring standard formulas for unbiased estimators (dividing by n-1 for variance) and a routine two-sample z-test for means with large samples. The central limit theorem part requires only recall of its role in justifying normality. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks |
|---|---|
| (a) \(\mu = \bar{x} = \frac{7335}{500} = 14.7\) | M1 A1 |
| \(\sigma^2 = s^2 = \frac{500}{499}(\frac{172040}{500} - 14.67^2) = 129.1\) | M2 A1 |
| (b) \(H_0: \mu_E = \mu_M\) \(H_1: \mu_E > \mu_M\) | B1 |
| 5% level \(\therefore\) C.R. is \(z > 1.6449\) | M1 A1 |
| test statistic \(= \frac{15.9-14.7}{\sqrt{\frac{185^2}{199}+\frac{181^2}{181}}} = 1.34\) | M2 A2 |
| Answer | Marks | Guidance |
|---|---|---|
| no evidence of difference in mean length of calls | M1 A1 | |
| (c) distributions not necessarily normal but by CLT sample mean distributed approximately normally whatever dist. for large sample \(\therefore\) can do test | B2 | (16) |
| Answer | Marks | Guidance |
|---|---|---|
| Total |
(a) $\mu = \bar{x} = \frac{7335}{500} = 14.7$ | M1 A1 |
$\sigma^2 = s^2 = \frac{500}{499}(\frac{172040}{500} - 14.67^2) = 129.1$ | M2 A1 |
(b) $H_0: \mu_E = \mu_M$ $H_1: \mu_E > \mu_M$ | B1 |
5% level $\therefore$ C.R. is $z > 1.6449$ | M1 A1 |
test statistic $= \frac{15.9-14.7}{\sqrt{\frac{185^2}{199}+\frac{181^2}{181}}} = 1.34$ | M2 A2 |
$1.34 < 1.6449 \therefore$ do not reject $H_0$
no evidence of difference in mean length of calls | M1 A1 |
(c) distributions not necessarily normal but by CLT sample mean distributed approximately normally whatever dist. for large sample $\therefore$ can do test | B2 | (16)
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**Total** | | | (75)7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price.
The company collected data on the time, $t$ minutes, of 500 calls made by young people on mobiles and the data is summarised by
$$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the mean and variance of $t$.
For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes ${ } ^ { 2 }$ respectively.
\item Stating your hypotheses clearly, test at the $5 \%$ level whether or not there is evidence that longer calls are made on land lines than on mobiles.\\
(9 marks)
\item Explain the importance of the central limit theorem in carrying out the test in part (b).
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q7 [16]}}