S2 June 2018 Q5

EdexcelOld spec16 marksIncludes hypothesis testingBinomial Distribution Normal Distribution

5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35

During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.

(a) Test, at the 5% level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. (5)

The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.

(b) Using a 5% level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to 2.5% (3)

During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.

(c) Using your answer to part (b), state whether or not this sample supports the manager’s claim. Use a 5% level of significance. (1)

(d) State the actual significance level of this test. (1)

The proportion of customers who buy organic fruit from Tesson supermarket is 0.2
During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than \(n\) of these customers bought organic fruit is 0.0465 correct to 4 decimal places.

(e) Find the value of \(n\). (6)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H}_0 : p = 0.35 \quad \mathrm{H}_1 : p \gt 0.35\)	B1
\(V \sim \mathrm{B}(40, 0.35)\) \(\mathrm{P}(V \geqslant 18) = 1 - \mathrm{P}(V \leqslant 17)\) \| or \(\mathrm{P}(V \geqslant 19) = 0.0699\)	M1
\(= 1 - 0.8761\) \| \(\mathrm{P}(V \geqslant 20) = 0.0363\) \(= 0.1239\) \| CR \(V \geqslant 20\)	A1
Accept \(\mathrm{H}_0\) or not Significant or 18 does not lie in the critical region	M1d
There is insufficient evidence that the proportion/amount/number/ percentage of customers who bought organic vegetables has increased.	A1cso
	(5)

Notes

B1 both hypotheses correct with \(p\) or \(\pi\)

M1 writing or using \(V \sim \mathrm{B}(40, 0.35)\) and \(1 - \mathrm{P}(V \leqslant 17)\) or \(\mathrm{P}(V \leqslant 17) = 0.8761\) or awrt 0.124 OR writing \(\mathrm{P}(V \geqslant 19) = 0.0699\) or \(\mathrm{P}(V \geqslant 20) = 0.0363\) leading to a CR. Implied by correct CR

A1 awrt 0.124 or \(V \geqslant 20\) or \(V \gt 19\) allow any letter

M1d dep on previous M being awarded. ft their CR or probability. A correct statement – do not allow contradicting non-contextual comments

A1 cso all previous marks must be awarded. A correct statement in context. Need Bold words. NB award M1A1 for a correct contextual statement on its own. If there are no hypotheses or they are the wrong way around, then M0A0

Mark scheme (b)

Scheme	Marks
\(E \sim \mathrm{B}(50, 0.35)\)	M1
\(\mathrm{P}(E \leqslant 10) = 0.0160 \qquad \mathrm{P}(E \geqslant 25) = 0.0207\) \(\mathrm{P}(E \leqslant 11) = 0.0342 \qquad \mathrm{P}(E \geqslant 24) = 0.0396\)
CR \(E \leqslant 10 \qquad E \geqslant 25\)	A1A1
	(3)

Notes

M1 writing \(E \sim \mathrm{B}(50, 0.35)\) or a correct probability or one tail of the CR correct

A1 \(E \leqslant 10\) oe A1 \(E \geqslant 25\) oe, allow any letter. Condone missing letter

NB If CR written as probabilities and both are correct or CR written as \(10 \geqslant E \geqslant 25\) oe award M1A1A0. If just give CV 10 and 25 given award M1A0A0

Mark scheme (c)

Scheme	Marks
The manager’s claim is supported or there is sufficient evidence that the proportion of customers buying organic eggs is different from those buying organic vegetables.	B1ft
	(1)

Notes

B1 A correct statement including the words managers claim or eggs and vegetable(s) (or veg) ft their 2 tail CR. Cannot be awarded if no CR given in (b)

Scheme	Marks
\(0.016 + 0.0207 = 0.0367\) or 3.67% awrt 0.0367 or 3.67%	B1
	(1)

Mark scheme (e)

Scheme	Marks
\(F \sim \mathrm{N}(40, 32)\)	M1 A1
\(\mathrm{P}(F \lt n) = \mathrm{P}\left(Z \lt \dfrac{n - 0.5 - 40}{\sqrt{32}}\right)\)	M1M1d
\(\dfrac{n - 0.5 - 40}{\sqrt{32}} = -1.68\)	B1
\(n = 31\)	A1cso
	(6)
	(16 marks)

Notes

M1 writing/using normal approximation with mean = 40

A1 writing/using normal approximation with mean = 40 and var = 32

M1 \(\pm\left(\dfrac{(n \text{ or } n - 0.5 \text{ or } n + 0.5) - \text{their mean}}{\text{their sd}}\right)\) if no mean or sd given they must be correct here.

M1 dep on previous method mark being awarded. Using continuity correction \(n - 0.5\)

B1 \(\pm\) 1.68 A1 31 cso all previous marks must be awarded.

NB 31 with no working gains no marks