S2 June 2006 Q7

EdexcelOld spec14 marksIncludes hypothesis testingBinomial Distribution

7. It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

(a) Using a 5% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5% as possible. (6)

(b) State the actual significance level of the above test. (1)

At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.

(c) Test, at the 10% level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly. (7)

Mark scheme (a)

Scheme	Marks
Let \(X\) represent the number of bowls with minor defects. \(\therefore X \sim \mathrm{B}; (25, 0.20)\)	B1; B1
\(\mathrm{P}(X \leqslant 1) = 0.0274 \quad\) or \(\mathrm{P}(X = 0) = 0.0038\)	M1A1
\(\mathrm{P}(X \leqslant 9) = 0.9827; \Rightarrow \mathrm{P}(X \geqslant 10) = 0.0173\)	A1
\(\therefore\) CR is \(\{X \leqslant 1 \cup X \geqslant 10\}\)	A1
	(6)

Notes

B1; B1 may be implied

M1A1 need to see at least one prob for \(X \leqslant\) no for M1

2nd A1 either

Scheme	Marks
Significance level \(= 0.0274 + 0.0173\) \(= 0.0447\) or 4.477%	B1
	(1)

Mark scheme (c)

Scheme	Marks
\(\mathrm{H}_0: p = 0.20;\ \ \mathrm{H}_1: p \lt 0.20;\)	B1 B1
Let \(Y\) represent number of bowls with minor defects Under \(\mathrm{H}_0\) \(Y \sim \mathrm{B}(20, 0.20)\)	B1
\(\mathrm{P}(Y \leqslant 2) \quad\) or \(\mathrm{P}(Y \leqslant 2) = 0.2061\) \(\mathrm{P}(Y \leqslant 1) = 0.0692\)	M1
\(= 0.2061 \qquad \text{CR } Y \leqslant 1\)	A1
\(0.2061 \gt 0.10\) or \(0.7939 \lt 0.9\) or \(2 \gt 1\)	M1
Insufficient evidence to suggest that the proportion of defective bowls has decreased.	B1ft
	(7)
	(14 marks)

Notes

3rd B1 may be implied

1st M1 either

2nd M1 their p