| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two-sample t-test (unknown variances) |
| Difficulty | Standard +0.3 This is a standard two-sample t-test with large sample sizes (n=100, 150), making the CLT application straightforward. Part (a) requires routine hypothesis test execution, part (b) tests understanding of why the test is valid, and part (c) asks for a standard assumption (independence). The large samples simplify calculations and remove concerns about normality, making this slightly easier than an average A-level question despite being S3 content. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(H_0: \mu_d = \mu_b\), \(H_1: \mu_d < \mu_b\) | B1 | |
| s.e. = \(\sqrt{\frac{25^2}{100} + \frac{10^2}{150}}\), \(z = \frac{67-60}{\sqrt{\frac{25^2}{100}+\frac{10^2}{150}}}\) | M1, dM1 | |
| \(CR = 1.6449 \times \sqrt{\frac{25^2}{100}+\frac{10^2}{150}}\) | ||
| \(z = \pm 2.6616....\) | ||
| One tailed critical value = 1.6449 (or prob of awrt 0.004 (<0.05)) | ||
| [Condone 0.996 if compared correctly with 0.95 for the B1] | ||
| 2.6616 > 1.6449 so significant evidence to reject \(H_0\) | A1 B1 | |
| There is evidence that the amount of lead present in the soil has decreased. | dM1 A1ft | (7) |
| (b) CLT enables you to assume that means are normally distributed | B1 | (1) |
| (c) Have assumed \(s^2 = \sigma^2\) or variance of sample = variance of population | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| 3rd dM1 dep. on 2nd M1 for a correct statement based on their normal cv (\( | cv | > 1.5\)) and their test statistic |
**(a)** $H_0: \mu_d = \mu_b$, $H_1: \mu_d < \mu_b$ | B1 |
s.e. = $\sqrt{\frac{25^2}{100} + \frac{10^2}{150}}$, $z = \frac{67-60}{\sqrt{\frac{25^2}{100}+\frac{10^2}{150}}}$ | M1, dM1 |
$CR = 1.6449 \times \sqrt{\frac{25^2}{100}+\frac{10^2}{150}}$ | |
$z = \pm 2.6616....$ | |
One tailed critical value = 1.6449 (or prob of awrt 0.004 (<0.05)) | |
[Condone 0.996 if compared correctly with 0.95 for the B1] | |
2.6616 > 1.6449 so significant evidence to reject $H_0$ | A1 B1 |
There is evidence that the amount of lead present in the soil has decreased. | dM1 A1ft | (7)
**(b)** CLT enables you to assume that means are normally distributed | B1 | (1)
**(c)** Have assumed $s^2 = \sigma^2$ or variance of sample = variance of population | B1 | (1)
**[Total 9]**
**Notes:**
**(a)** 1st B1 for both hypotheses in terms of $\mu$ not words. Accept $\mu_1, \mu_2$ etc if there is some indication of which is which e.g. X ~ N(67, $25^2$) implies X is "before".
1st M1 for attempt at s.e. - condone one number wrong or mis-matched variances i.e. $\sqrt{\frac{p}{q}+\frac{r}{s}}$ (3 of p,q,r & s correct) or $\sqrt{\frac{100 + 150}{}}$
2nd dM1 Dep on 1st M1 for using their s.e. in incorrect formula for test statistic. Num of $\pm (67 - 60)$ or for correct expression for CR
3rd dM1 dep. on 2nd M1 for a correct statement based on their normal cv ($|cv| > 1.5$) and their test statistic
2nd A1ft for correct comment in context. Must mention "lead" or "soil" and "factory". Allow ft. If hypotheses are the wrong way round score A0
**(b)** B1 must mention mean and normal. In words or symbols e.g. $\bar{X} \sim N(...$
---7. A farmer monitored the amount of lead in soil in a field next to a factory.
He took 100 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be $67 \mathrm { mg } / \mathrm { kg }$ with standard deviation $25 \mathrm { mg } / \mathrm { kg }$.\\
After the factory closed, the farmer took 150 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be $60 \mathrm { mg } / \mathrm { kg }$ with standard deviation $10 \mathrm { mg } / \mathrm { kg }$.
\begin{enumerate}[label=(\alph*)]
\item Test at the $5 \%$ level of significance whether or not the mean weight of lead in the soil decreased after the factory closed. State your hypotheses clearly.
\item Explain the significance of the Central Limit Theorem to the test in part(a).
\item State an assumption you have made to carry out this test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2013 Q7 [9]}}