S4 January 2006 Q6

EdexcelOld spec12 marksConfidence Intervals

6. A tree is cut down and sawn into pieces. Half of the pieces are stored outside and half of the pieces are stored inside. After a year, a random sample of pieces is taken from each location and the hardness is measured. The hardness \(x\) units are summarised in the following table.

	Number of pieces sampled	\(\Sigma x\)	\(\Sigma x^2\)
Stored outside	20	2340	274050
Stored inside	37	4884	645282

(a) Show that unbiased estimates for the variance of the values of hardness for wood stored outside and for the wood stored inside are 14.2 and 16.5, to 1 decimal place, respectively. (2)

The hardness of wood stored outside and the hardness of wood stored inside can be assumed to be normally distributed with equal variances.

(b) Calculate 95% confidence limits for the difference in mean hardness between the wood that was stored outside and the wood that was stored inside. (8)

(c) Using your answer to part (b), comment on the means of the hardness of wood stored outside and inside. Give a reason for your answer. (2)

Mark scheme (a)

Scheme	Marks
\(\hat{\sigma}^2_{\text{outside}} = \dfrac{1}{19}\left(274050 - \dfrac{(2340)^2}{20}\right) = 14.2\)	M1
\(\hat{\sigma}^2_{\text{inside}} = \dfrac{1}{36}\left(645282 - \dfrac{(4884)^2}{37}\right) = 16.5\) * AG both	A1
	(2)

Notes

(corrected from the printed mark scheme: the scheme labels these two estimates \(\sigma_{\mathrm{I}}\) and \(\sigma_{\mathrm{o}}\), with \(\sigma_{\mathrm{I}}\) on the stored-outside value; they are the variance estimates for wood stored outside and inside respectively)

Mark scheme (b)

Scheme	Marks
\(s_p = \sqrt{\dfrac{19 \times 14.2 + 36 \times 16.5}{55}} = \sqrt{15.705} = 3.963\ldots\)	M1 A1
Mean outside \(= \dfrac{2340}{20} = 117\), Mean inside \(= 132\)	B1 B1
Confidence limits \(= (132 - 117) \pm 2.004 \times 3.963\ldots\sqrt{\dfrac{1}{20} + \dfrac{1}{37}}\)	M1 A1ft
\(= (12.8,\ 17.2)\)	A1 A1
	(8)