S2 June 2013 Q6

EdexcelOld spec10 marksIncludes hypothesis testingBinomial Distribution

6. In a manufacturing process 25% of articles are thought to be defective. Articles are produced in batches of 20

(a) A batch is selected at random. Using a 5% significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25
You should state the probability in each tail which should be as close as possible to 0.025 (5)

The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.

(b) Test at the 5% level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly. (5)

Mark scheme (a)

Scheme	Marks
\(X \sim \mathrm{B}(20, 0.25)\)	M1
\(\mathrm{P}(X \geqslant 10) = 1 - 0.9861 = 0.0139\)	A1
\(\mathrm{P}(X \leqslant 1) = 0.0243\)	A1
\((0 \leqslant)X \leqslant 1 \cup 10 \leqslant X(\leqslant 20)\)	A1A1
	(5)

Notes

M1 using B(20,0.25) may be implied by a correct CR (allow written as a probability statement)

1st A1 awrt 0.0139

2nd A1 awrt 0.0243

3rd A1 \(X \leqslant 1\) or \(0 \leqslant X \leqslant 1\) or [0,1] or 0,1 or equivalent statements

4th A1 \(X \geqslant 10\) or \(10 \leqslant X \leqslant 20\) or 10,11,12,13,14,15,16,17,18,19,20 or [10,20] or equivalent statements

NB These two A marks must be for statements with \(X\) (any letter) only – not in probability statements and SC for CR written as \(1 \geqslant X \geqslant 10\) gets A1 A0

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : p = 0.25\) \(\mathrm{H}_1 : p \lt 0.25\)	B1
\(X \sim \mathrm{B}(20, 0.25)\) \(\mathrm{P}(X \leqslant 3) = 0.2252\) or CR \(X \leqslant 1\)	M1A1
Insufficient evidence to reject \(\mathrm{H}_0\), Accept \(\mathrm{H}_0\), Not significant. 3 does not lie in the Critical region.	M1d
No evidence that the changes to the process have reduced the percentage of defective articles (oe)	A1cso
	(5)
	(10 marks)

Notes

B1 both hypotheses with \(p\)

1st M1 using B(20, 0.25) and finding \(\mathrm{P}(X \leqslant 3)\) or \(\mathrm{P}(X \geqslant 4)\) may be implied by a correct CR

1st A1 0.2252 (allow 0.7748) if not using CR or CR \(X \leqslant 1\) or \(X \lt 2\)

2nd M1 dependent on previous M being awarded. A correct statement (do not allow if there are contradicting non contextual statements)

A1cso Conclusion must contain the words changes/new process oe, reduced oe number/percentage oe , and defective articles/defectives. There must be no incorrect working seen.