S4 June 2013 Q3

EdexcelOld spec12 marksIncludes hypothesis testingt-Tests & Further Tests

3. An archaeologist is studying the compression strength of bricks at some ancient European sites. He took random samples from two sites \(A\) and \(B\) and recorded the compression strength of these bricks in appropriate units. The results are summarised below.

Site	Sample size (\(n\))	Sample mean (\(\bar{x}\))	Standard deviation (\(s\))
\(A\)	7	8.43	4.24
\(B\)	13	14.31	4.37

It can be assumed that the compression strength of bricks is normally distributed.

(a) Test, at the 2% level of significance, whether or not there is evidence of a difference in the variances of compression strength of the bricks between these two sites. State your hypotheses clearly. (5)

Site \(A\) is older than site \(B\) and the archaeologist claims that the mean compression strength of the bricks was greater at the younger site.

(b) Stating your hypotheses clearly and using a 1% level of significance, test the archaeologist’s claim. (6)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H}_0 : \sigma_A^2 = \sigma_B^2 \qquad \mathrm{H}_1 : \sigma_A^2 \neq \sigma_B^2\)	B1
\(F = \dfrac{s_B^{\,2}}{s_A^{\,2}} = \dfrac{4.37^2}{4.24^2} = 1.0622\ldots\)	M1A1
\(F_{12,6}\ (0.01) = 7.72\)	B1
Not sig, so no evidence of a difference in variances	A1ft
	(5)

Notes

M1 for use of a correct formula Allow \(F = \dfrac{s_A^{\,2}}{s_B^{\,2}} = \dfrac{4.24^2}{4.37^2} = 0.941\ldots\) with 0.1295..

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \mu_A = \mu_B \qquad \mathrm{H}_1 : \mu_A \lt \mu_B\)	B1
\(s_p^{\,2} = \dfrac{6 \times 4.24^2 + 12 \times 4.37^2}{18} = 18.7238\) or \(s_p = 4.327\ldots\)	M1
\(t = \pm\dfrac{14.31 - 8.43}{s_p\sqrt{\dfrac{1}{7} + \dfrac{1}{13}}} = \pm 2.8985\ldots\) awrt 2.9	M1A1
\(t_{18}(0.01) = 2.552\)	B1
sig, there is evidence to support archaeologist’s claim or there is evidence that bricks for site \(B\) have higher mean compression strength than those from site \(A\).	A1ft
	(6)

Notes

B1 if \(A\) and \(B\) not used it must be clear which is \(A\) and which is \(B\)
1^st M1 for attempt to calculate \(s_p\) or \(s_p^{\,2}\)
2^nd M1 for attempt correct test statistic
2^nd A1 ft need archaeologist’s or compression

Mark scheme (c)

Scheme	Marks
The test in (b) requires \(\sigma_A^2 = \sigma_B^2\) and the test in part (a) shows that this is a reasonable assumption. (o.e.)	B1
	(1)
	(12 marks)

Notes

Need to refer to ‘allows us to assume variances the same’ and this is needed in for test. oe