S3 June 2013 (R) Q7

EdexcelOld spec9 marksIncludes hypothesis testingt-Tests & Further Tests

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 mg/kg with standard deviation 25 mg/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 mg/kg with standard deviation 10 mg/kg.

(a) 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. (7)

(b) Explain the significance of the Central Limit Theorem to the test in part (a). (1)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H_0}: \mu_a = \mu_b\), \(\mathrm{H_1}: \mu_a \lt \mu_b\)	B1
s.e. \(= \sqrt{\dfrac{25^2}{100} + \dfrac{10^2}{150}}\), \(z = \dfrac{67 - 60}{\sqrt{\dfrac{25^2}{100} + \dfrac{10^2}{150}}}\) CR \(= 1.6449 \times \sqrt{\dfrac{25^2}{100} + \dfrac{10^2}{150}}\)	M1,dM1
\(z = \pm 2.6616\ldots\) \(= \pm 4.326\ldots\) (awrt 2.66/4.33)	A1
One tailed critical value \(z = 1.6449\) (or prob of awrt 0.004 (<0.05)) [Condone 0.996 if compared correctly with 0.95 for the B1]	B1
[2.6616 > 1.6449 so] significant evidence to reject \(\mathrm{H_0}\)	dM1
There is evidence that the amount of lead present in the soil has decreased.	A1ft
	(7)

Notes

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 \(X \sim \mathrm{N}(67, 25^2)\) 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 \(\sqrt{\frac{10^2}{100} + \frac{25^2}{150}}\)

2nd dM1 Dep on 1st M1 for using their s.e. in correct 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 (\(|\text{cv}| \gt 1.5\)) 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
If hypotheses are not for a difference between 2 means award A0

Scheme	Marks
CLT enables you to assume that means are normally distributed	B1
	(1)

Scheme	Marks
Have assumed \(s^2 = \sigma^2\) or variance of sample = variance of population	B1
	(1)
	(9 marks)