S2 June 2017 Q1

EdexcelOld spec9 marksIncludes hypothesis testingBinomial Distribution

1. A potter believes that 20% of pots break whilst being fired in a kiln. Pots are fired in batches of 25.

(a) Let \(X\) denote the number of broken pots in a batch. A batch is selected at random. Using a 10% significance level, find the critical region for a two tailed test of the potter’s belief. You should state the probability in each tail of your critical region. (4)

The potter aims to reduce the proportion of pots which break in the kiln by increasing the size of the batch fired. He now fires pots in batches of 50. He then chooses a batch at random and discovers there are 6 pots which broke whilst being fired in the kiln.

(b) Test, at the 5% level of significance, whether or not there is evidence that increasing the number of pots in a batch has reduced the percentage of pots that break whilst being fired in the kiln. State your hypotheses clearly. (5)

Mark scheme (a)

Allow any letter instead of \(X\) or \(c\) for this question

Scheme	Marks
\(X \sim \mathrm{B}(25, 0.2)\)	M1
\(\left[\mathrm{P}(X \geqslant 9) =\right] 0.0468\) \(\left[\mathrm{P}(X \leqslant 1) =\right] 0.0274\)	A1
\(X = [0 \leqslant]\ X \leqslant 1\)	A1
\(9 \leqslant X\ [\leqslant 25]\)	A1d
	(4)

Notes

M1 Writing or using B(25, 0.2) or B(25, 1/5) [allow Po(5)] May be written in full or implied by a correct CR (allow written as a probability statement)

1st A1 both awrt 0.0468 and awrt 0.0274 seen.

2nd A1 \(X \leqslant 1\) or \(X \lt 2\) or \(0 \leqslant X \leqslant 1\) or [0,1] or 0,1 or equivalent statements. \(X \leqslant c\) and \(c = 1\)

3rd A1d dependent on seeing a probability from the B(25, 0.2) and \(X \geqslant 9\) or \(X \gt 8\) or \(9 \leqslant X \leqslant 25\) or 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 or [9,25] or equivalent statements. \(X \geqslant c\) and \(c = 9\)

NB These two final 2 A marks must be for statements with “\(X\)” only(or list) – not in probability statements

SC If a probability from the B(25, 0.2) is seen and they either have both CR correct but written as probability statements or the CR is written as \(1 \geqslant X \geqslant 9\) they get A1 A0 for final 2 marks

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : p = 0.2\) \(\mathrm{H}_1 : p \lt 0.2\)	B1
\(\mathrm{P}(X \leqslant 6) = 0.1034\) or CR \(X \leqslant 5\)	M1 A1
Insufficient evidence to reject \(\mathrm{H}_0\), Accept \(\mathrm{H}_0\), Not significant. 6 does not lie in the Critical region.	M1d
No evidence that increasing the batch size has reduced the percentage of broken pots (oe) or evidence that there is no change in the percentage of broken pots (oe)	A1cso
	(5)
	(9 marks)

Notes

B1 both hypotheses with \(p\) or \(\pi\) and clear which is \(\mathrm{H}_0\) and which is \(\mathrm{H}_1\)

1st M1 writing or using B(50, 0.2) and writing or using \(\mathrm{P}(X \leqslant 6)\) or \(\mathrm{P}(X \geqslant 7)\) on its own. May be implied by a correct CR

1st A1 awrt 0.103. Allow CR \(X \leqslant 5\) or \(X \lt 6\). or if not using CR allow awrt 0.897.

2nd M1 dependent on previous M being awarded. A correct statement (do not allow if there are contradicting non-contextual statements). ft their Prob/CR compared with 0.05/6/(0.95 if using 0.8979). Do not follow through their hypotheses

2nd A1cso Conclusion must contain the words reduced/ no change/not affect oe number/percentage/proportion/ probability oe, and pots. All previous marks must be awarded for this mark to be awarded.
Do not allow the potters claim /belief is wrong/true
NB Correct contextual statement on its own scores M1A1