S2 June 2005 Q7

EdexcelOld spec14 marksIncludes hypothesis testingBinomial Distribution

7. A drugs company claims that 75% of patients suffering from depression recover when treated with a new drug.

A random sample of 10 patients with depression is taken from a doctor’s records.

(a) Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. (2)

Given that the claim is correct,

(b) find the probability that the treatment will be successful for exactly 6 patients. (2)

The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.

(d) From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the 1% level. (4)

Scheme	Marks
\(X \sim \mathrm{B}(10, p)\)	B1, B1
	(2)

Scheme	Marks
\(\mathrm{P}(X = 6) = 0.9219 - 0.7759\)	M1
\(= 0.1460\)	A1
	(2)

Mark scheme (c)

Scheme	Marks
\(\mathrm{H}_0: p = 0.75\) (or \(p = 0.25\))	B1
\(\mathrm{H}_1: p \lt 0.75\) (or \(p \gt 0.25\))	B1
Under \(\mathrm{H}_0\), \(X \sim \mathrm{B}(20, 0.75)\) (or \(Y \sim \mathrm{B}(20, 0.25)\))	B1
\(\mathrm{P}(X \leqslant 13) = 1 - 0.7858 = 0.2142\) (or \(\mathrm{P}(Y \geqslant 7)\)) Insufficient evidence to reject \(\mathrm{H}_0\) as \(0.2142 \gt 0.05\)	M1, A1
Doctor’s belief is not supported by the sample	A1
	(6)

Notes

1st B1 correct \(\mathrm{H}_0\)

2nd B1 one tailed \(\mathrm{H}_1\)

3rd B1 implied

M1, A1 \(\mathrm{P}(X \leqslant 13)\) and 1 –, 0.2142

A1 context

(corrected from the printed mark scheme: the printed scheme has “0.2412 > 0.05”; the probability found is 0.2142)

Alternative (critical region)

Scheme	Marks
(OR CR \(\mathrm{P}(X \leqslant 12) = 1 - 0.8982 = 0.1018\) (or \(\mathrm{P}(Y \geqslant 8)\)) \(\mathrm{P}(X \leqslant 11) = 1 - 0.9591 = 0.0409\) (or \(\mathrm{P}(Y \geqslant 9)\)) 13 outside critical region (or 7))	(M1 A1)

(M1 A1) either

Mark scheme (d)

Scheme	Marks
\(\mathrm{P}(X \leqslant c) \leqslant 0.01\) for \(p = 0.75\) (or \(\mathrm{P}(Y \geqslant 20 - c) \leqslant 0.01\) for \(p = 0.25\))	M1 A1
\(\mathrm{P}(X \leqslant 9) = 1 - 0.9961 = 0.0039\) (or \(\mathrm{P}(Y \geqslant 11)\)) \(\mathrm{P}(X \leqslant 10) = 1 - 0.9861 = 0.0139\) (or \(\mathrm{P}(Y \geqslant 10)\))	B1
C. R. is [0,9], so greatest no. of patients is 9.	B1
	(4)
	(14 marks)

Notes

1st B1 0.9961 or 0.9981

2nd B1 9