S4 June 2013 Q2

EdexcelOld spec10 marksIncludes hypothesis testingConfidence Intervals t-Tests & Further Tests

2. Every 6 months some engineers are tested to see if their times, in minutes, to assemble a particular component have changed. The times taken to assemble the component are normally distributed. A random sample of 8 engineers was chosen and their times to assemble the component were recorded in January and in July. The data are given in the table below.

Engineer	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
January	17	19	22	26	15	28	18	21
July	19	18	25	24	17	25	16	19

(a) Calculate a 95% confidence interval for the mean difference in times. (7)

(b) Use your confidence interval to state, giving a reason, whether or not there is evidence of a change in the mean time to assemble a component. State your hypotheses clearly. (3)

Mark scheme (a)

Scheme	Marks
\(d\) = Jan - July: −2, 1, −3, 2, −2, 3, 2, 2	M1
\(\bar{d} = 0.375,\quad \sum d^2 = 39 \Rightarrow s^2 = 5.4107\ldots\) or \(s = 2.326\ldots\)	M1, M1
\(t_7(0.025) = 2.365\)	B1
Confidence Interval: \(0.375 \pm 2.365 \times \dfrac{2.326\ldots}{\sqrt{8}}\)	M1
\(= \underline{\mathbf{(-1.57,\ 2.32)}}\) (o.e.)	A1,A1
	(7)

Notes

1^st M1 for attempting differences
2^nd M1 for attempting \(\bar{d}\)
3^rd M1 for attempting \(s_d^{\,2}\), correct expression with their \(\sum d^2\) and \(\bar{d}\) or correct calculation (to 2 sf or better)
4^th M1 for use of a correct CI formula, using a value for \(t\) and ft their values.
1^st A1 for lower limit of -1.57 or -2.32
2^nd A1 for corresponding upper limit

S.C. Allow A1A1 for (0, 2.32)

(corrected from the printed mark scheme: the first line is printed as “d = Jan - June”; the second set of times is July) CHECK

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \mu_D = 0 \qquad \mathrm{H}_1 : \mu_D \neq 0\)	B1
Comment that 0 is in the interval	M1
Not sig, no evidence of a change in mean time to assemble component	A1ft
	(3)
	(10 marks)

Notes

B1 for both hypotheses using \(\mu_D\)
M1 for a comment about 0 being in (or out) of their interval
A1 contextual conclusion – must include assemble components

S.C. If they have used difference in means test in part (a) to get the confidence interval then award the B1 for \(\mathrm{H}_0 : \mu_x - \mu_y = 0 \quad \mathrm{H}_1 : \mu_x - \mu_y \neq 0\) or the correct hypotheses.