A2 June 2019 Q1

EdexcelCurrent spec8 marksConfidence Intervals

1. A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams.

To check that the machine is working properly, a random sample of 8 pots is selected.
The weight of yoghurt, in grams, in each pot is as follows

\[508 \qquad 510 \qquad 500 \qquad 500 \qquad 498 \qquad 503 \qquad 508 \qquad 505\]

Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,

(a) find a 95% confidence interval for the mean weight, \(\mu\) grams, of yoghurt in a pot.
Give your answers to 2 decimal places. (4)

(b) Comment on whether or not the machine is working properly, giving a reason for your answer. (1)

(d) Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer.
You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution. (2)

Mark scheme (a)

Scheme	Marks	AO
Mean = 504	B1	1.1b
1.96	B1	3.3
\(504 \pm \dfrac{5.4}{\sqrt{8}} \times \text{“}1.96\text{”}\)	M1	2.1
(500.258, 507.742)	A1	1.1b
	(4)

Notes

B1: 504 may be seen in part (b)

B1: For realising a normal distribution must be used as a model and finding the correct value 1.96

M1: For \(504 \pm \dfrac{5.4}{\sqrt{8}} \times \text{“}z\text{ value}\text{”}\). \(|z| \gt 1\) May be implied by a correct CI

A1: awrt 500.26 and 507.74 NB using \(t\) gives 500.29 and 507.71

Mark scheme (b)

Scheme	Marks	AO
505 is in the confidence interval therefore there is evidence that the machine is working properly	B1ft	2.2b
	(1)

Notes

B1ft: Drawing a correct inference (ft) using their answer to part (a) and the 505 from the question. Reason must be given. Ignore incorrect non – contextual

Scheme	Marks	AO
5% oe	B1	1.1b
	(1)

Mark scheme (d)

Scheme	Marks	AO
\(s\) needs to be used instead of \(\sigma\) and a \(t\)-value instead of the \(z\) value	B1	3.3
since the sample is small therefore you can’t use the normal distribution	B1	3.5b
	(2)
	(8 marks)

Notes

B1: create new model by using \(s\) and \(t\). Allow if state use CI \(\mu \pm \dfrac{s}{\sqrt{n}} \times \text{“}t\text{”}\) or use \(s = 4.44\) and \(t = 2.365\)

B1: For recognising that the sample is small