A2 June 2023 Q2

EdexcelCurrent spec12 marksIncludes hypothesis testingConfidence Intervals t-Tests & Further Tests

2. Camilo grows two types of apple, green apples and red apples.

The standard deviation of the weights of green apples is known to be 3.5 grams.

A random sample of 80 green apples has a mean weight of 128 grams.

(a) Find a 98% confidence interval for the mean weight of the population of green apples.
Show your working clearly and give the confidence interval limits to 2 decimal places. (3)

Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples.

A random sample of \(n\) red apples has a mean weight of 117 grams.

The standard deviation of the weights of the red apples is known to be 4 grams.

A test of Camilo’s belief is carried out at the 5% level of significance.

(b) State the null and alternative hypotheses for this test. (1)

(d) Explain the relevance of the Central Limit Theorem in parts (a) and (c). (1)

(e) Given that \(n = 85\), state the conclusion of the hypothesis test. (1)

Mark scheme (a)

Scheme	Marks	AO
\(z = 2.3263\)	B1	1.1b
98% CI for \(\mu\) is: \(128 \pm 2.3263 \times \dfrac{3.5}{\sqrt{80}}\)	M1	2.1
\(= (127.0896\ldots,\ 128.9103\ldots) = \) awrt \((127.09,\ 128.91)\)	A1	1.1b
	(3)

Notes

B1: 2.3263 or better

M1: for \(128 \pm \sqrt{\dfrac{3.5^2}{80}} \times \text{“}z\ \text{value}\text{”}\). \(|z| \gt 1\) May be implied by a correct CI

A1: awrt 127.09 and awrt 128.91

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{H}_0 : \mu_{g(reen)} - \mu_{r(ed)} = 10 \qquad \mathrm{H}_1 : \mu_{g(reen)} - \mu_{r(ed)} \gt 10\)	B1	2.5
	(1)

Notes

B1: Both hypotheses (oe) correct with correct notation (if using \(\mu_x\) and \(\mu_y\) these must be defined)

Mark scheme (c)

Scheme	Marks	AO
Normal model: \(\bar{G} - \bar{R} \sim \mathrm{N}\left(10,\ \dfrac{3.5^2}{80} + \dfrac{4^2}{n}\right)\)	M1 A1	3.3 1.1b
\(z = \dfrac{(128 - 117) - 10}{\sqrt{\dfrac{3.5^2}{80} + \dfrac{4^2}{n}}} \gt 1.6449\)	M1 B1	3.1b 1.1b
\(\dfrac{1}{1.6449} \gt \sqrt{\dfrac{3.5^2}{80} + \dfrac{4^2}{n}} \to \left(\dfrac{1}{1.6449}\right)^2 - \dfrac{12.25}{80} \gt \dfrac{16}{n} \to n \gt 73.9\ldots\)	M1	1.1b
\(n = 74\)	A1	1.1b
	(6)

Notes

M1: setting up normal model with correct mean or variance
May be shown in standardisation with correct mean or variance and \(|z| \gt 1\)

A1: correct variance and mean, may be seen in standardisation

M1: setting up an equation or inequality by standardising with their mean and their standard deviation and \(|z| \gt 1\)

B1: Correct critical value 1.6449 or better, allow 1.64 or better

M1: rearranging to solve for \(n\) (may be implied by 73.9…)

A1: \(n = 74\) cao

Mark scheme (d)

Scheme	Marks	AO
Sample sizes are large so CLT means even though we do not know the distributions of \(G\) and \(R\), the means \(\bar{G}\) and \(\bar{G} - \bar{R}\) will be (approximately) normally distributed.	B1	2.4
	(1)

Notes

B1: correct explanation mentioning large sample size and the means being distributed normally

Mark scheme (e)

Scheme	Marks	AO
[Since 85 > 73.9…] there is significant evidence to support Camilo’s belief/the mean weight of green apples is more than 10 grams greater than red apples is supported.	B1ft	2.2b
	(1)
	(12 marks)

Notes

B1ft: drawing a correct inference in context, must be consistent with their value of \(n\)