S4 June 2006 Q4

EdexcelOld spec13 marksIncludes hypothesis testingConfidence Intervals t-Tests & Further Tests

4. Two machines \(A\) and \(B\) produce the same type of component in a factory. The factory manager wishes to know whether the lengths, \(x\) cm, of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below.

	Sample size	Mean \(\bar{x}\)	Standard deviation \(s\)
Machine \(A\)	9	4.83	0.721
Machine \(B\)	10	4.85	0.572

The lengths of components produced by the machines can be assumed to follow normal distributions.

(a) Use a two tail test to show, at the 10% significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal. (4)

(b) Showing your working clearly, find a 95% confidence interval for \(\mu_B - \mu_A\), where \(\mu_A\) and \(\mu_B\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively. (7)

There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm.

Mark scheme (a)

Scheme	Marks
\(\left(\mathrm{H}_0 : \sigma_A^2 = \sigma_B^2 \qquad \mathrm{H}_1 : \sigma_A^2 \neq \sigma_B^2\right)\) \(\dfrac{s_A^2}{s_B^2} = \dfrac{0.721^2}{0.572^2} = 1.588\ldots\) awrt 1.59	M1 A1
\(F_{8,9}\) (5%) c.v. [= 10% 2 tail] \(= 3.23\)	B1
Not significant, can assume variances are equal. (accept \(\sigma_A^2 = \sigma_B^2\))	B1 cao
	(4)

Mark scheme (b)

Scheme	Marks
\(s_p^2 = \dfrac{8 \times 0.721^2 + 9 \times 0.572^2}{8 + 9} = 0.41784\ldots\) \(0.417\ldots\) or awrt 0.418	M1 A1
\(t_{17}(2.5\%)\) c.v. \(= 2.110\)	B1
95% C.I. \(= \bar{x}_B - \bar{x}_A \pm 2.110 \times s_p \times \sqrt{\dfrac{1}{9} + \dfrac{1}{10}}\)	M1
\(= 0.02 \pm 2.110 \times \sqrt{0.417\ldots} \times \sqrt{\dfrac{1}{9} + \dfrac{1}{10}}\)	A1ft
\(= (-0.6066\ldots,\ 0.6466\ldots)\) awrt (−0.607, 0.647)	A1, A1
	(7)

Scheme	Marks
\(\pm 0.7\) is outside interval	B1ft
\(\therefore\) manager need not be concerned (dep)	B1ft
	(2)
	(13 marks)

Notes