S4 June 2014 (R) Q1

EdexcelOld spec9 marksIncludes hypothesis testingt-Tests & Further Tests

1. In a trial for a new cough medicine, a random sample of 8 healthy patients were given steadily increasing doses of a pepper extract until they started coughing. The level of pepper that triggered the coughing was recorded. Each patient completed the trial after taking a standard cough medicine and, at a later time, after taking the new medicine. The results are given in the table below.

	Level of pepper extract that triggers coughing
Patient	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Standard medicine	46	12	18	31	23	16	27	9
New medicine	53	16	13	49	11	34	38	22

(a) Using a suitable test, at the 5% level of significance, state whether or not, on the basis of this trial, you would recommend using the new medicine. State your hypotheses clearly. (8)

(b) State an assumption needed to carry out this test. (1)

Mark scheme (a)

Scheme	Marks
[New − standard = ] \(d\): 7, 4, −5, 18, −12, 18, 11, 13.	M1
\(\bar{d} = 6.75\)	M1
\({s_d}^2 = \dfrac{1172 - 8 \times 6.75^2}{7} = 115.3571\ldots\) or \(s_d = 10.7404\ldots\)	M1
\(\mathrm{H}_0 : \mu_d = 0 \qquad \mathrm{H}_1 : \mu_d \gt 0\)	B1
\(t_7 = \dfrac{6.75}{s_d/\sqrt{8}} = 1.7775\ldots\) or \(\dfrac{c}{s_d/\sqrt{8}} = 1.895\ \therefore\ \text{CR } c \gt \text{awrt } 7.2\) awrt 1.78	M1 A1
\(t_7(5\%)\) one tail critical value is 1.895 (or prob. = 0.05935…)	B1
Not significant. There is insufficient evidence that the new medicine is better or the new medicine is not recommended.	A1ft
	(8)

Notes

1st M1 for attempting the \(d\)s
2nd M1 for attempting \(\bar{d}\)
3rd M1 for attempting \(s_d\) or \({s_d}^2\)
1st B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\)

4th M1 for attempting the correct test statistic \(\dfrac{6.75}{s_d/\sqrt{8}}\) or \(p = \text{awrt } 0.06\) or \(\dfrac{c}{10.7/\sqrt{8}} = t\) value

1st A1 1.78 or awrt 0.06 or awrt 7.2
2nd B1 1.895 or awrt 0.06
2nd A1ft for a correct comment in context based on their test statistic and their cv.

Mark scheme (b)

Scheme	Marks
Need the differences between levels triggering coughing to be normally distributed	B1
	(1)
	(9 marks)

Notes

B1 for a comment that mentions “differences” and “normal” distribution