S2 January 2007 Q6

EdexcelOld spec13 marksIncludes hypothesis testingBinomial Distribution

6. Past records from a large supermarket show that 20% of people who buy chocolate bars buy the family size bar. On one particular day a random sample of 30 people was taken from those that had bought chocolate bars and 2 of them were found to have bought a family size bar.

(a) Test at the 5% significance level, whether or not the proportion \(p\), of people who bought a family size bar of chocolate that day had decreased. State your hypotheses clearly. (6)

The manager of the supermarket thinks that the probability of a person buying a gigantic chocolate bar is only 0.02. To test whether this hypothesis is true the manager decides to take a random sample of 200 people who bought chocolate bars.

(b) Find the critical region that would enable the manager to test whether or not there is evidence that the probability is different from 0.02. The probability of each tail should be as close to 2.5% as possible. (6)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H}_0: p = 0.20,\ \mathrm{H}_1: p \lt 0.20\)	B1,B1
Let \(X\) represent the number of people buying family size bar. \(X \sim \mathrm{B}(30, 0.20)\)
\(\mathrm{P}(X \leqslant 2) = 0.0442 \quad\) or \(\mathrm{P}(X \leqslant 2) = 0.0442\) \(\mathrm{P}(X \leqslant 3) = 0.1227\) \(\text{CR } X \leqslant 2\)	M1A1
\(0.0442 \lt 5\%\), so significant. Significant	M1
There is evidence that the no. of family size bars sold is lower than usual.	A1
	(6)

Notes

1st A1 awrt 0.044

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0: p = 0.02,\ \mathrm{H}_1: p \neq 0.02\)	B1
Let \(Y\) represent the number of gigantic bars sold. \(Y \sim \mathrm{B}(200, 0.02) \Rightarrow Y \sim \mathrm{Po}(4)\)	M1
\(\mathrm{P}(Y = 0) = \mathbf{0.0183}\) and \(\mathrm{P}(Y \leqslant 8) = \mathbf{0.9786} \Rightarrow \mathrm{P}(Y \geqslant 9) = \mathbf{0.0214}\)	B1,B1
Critical region \(Y = 0 \cup Y \geqslant 9\)	B1,B1
N.B. Accept exact Bin: 0.0176 and 0.0202

Notes

1st B1 \(\lambda = 4\) etc ok both

M1 can be implied below

B1,B1 first, either

B1,B1 \(Y \leqslant 0\) ok

Scheme	Marks
Significance level \(= 0.0183 + 0.0214 = 0.0397\)	B1
	(1)
	(13 marks)