S4 June 2016 Q1

EdexcelOld spec9 marksIncludes hypothesis testingt-Tests & Further Tests

1. A new diet has been designed. Its designers claim that following the diet for a month will result in a mean weight loss of more than 2 kg. In a trial, a random sample of 10 people followed the new diet for a month. Their weights, in kg, before starting the diet and their weights after following the diet for a month were recorded. The results are given in the table below.

Person	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Weight before diet (kg)	96	110	116	98	121	91	98	106	110	116
Weight after diet (kg)	91	101	111	96	121	91	90	101	104	110

(a) Using a suitable \(t\)-test, at the 5% level of significance, state whether or not the trial supports the designers’ claim. State your hypotheses and show your working clearly. (8)

(b) State an assumption necessary for the test in part (a). (1)

Mark scheme (a)

Scheme	Marks
\(d\): 5 9 5 2 0 0 8 5 6 6	M1
\(\bar{d} = \frac{\sum d}{n} = 4.6\)	M1
\(s^2 = \frac{296 - 10 \times 4.6^2}{9} = 9.378\)	M1
\(\mathrm{H}_0 : \mu_d = 2 \qquad \mathrm{H}_1 : \mu_d \gt 2\)	B1
\(t = \pm\dfrac{4.6 - 2}{\sqrt{\dfrac{9.378}{10}}} = \pm 2.6848\)	M1 A1
\(t_9(5\%) = \pm 1.833\ldots\)	B1
There is evidence to reject \(\mathrm{H}_0\). There is sufficient evidence to support the designers claim.	A1ft
	(8)

Notes

M1 for attempting the \(d\)s
M1 for attempting \(\bar{d}\)
M1 for \(s_d\) or \({s_d}^2\)
B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\). (allow a defined symbol)

M1 for attempting the correct test statistic \(\dfrac{\bar{d}}{s_d/\sqrt{10}}\)

A1 awrt 2.68
B1 awrt 1.83
A1ft for a correct comment in context

Scheme	Marks
The differences in weights are normally distributed.	B1
	(1)
	(9 marks)