A2 October 2021 Q7

EdexcelCurrent spec8 marksIncludes hypothesis testingQuality of Tests

7. A manufacturer has a machine that produces lollipop sticks.
The length of a lollipop stick produced by the machine is normally distributed with unknown mean \(\mu\) and standard deviation 0.2

Farhan believes that the machine is not working properly and the mean length of the lollipop sticks has decreased.
He takes a random sample of size \(n\) to test, at the 1% level of significance, the hypotheses

\[\mathrm{H}_0: \mu = 15 \qquad \mathrm{H}_1: \mu \lt 15\]

(a) Write down the size of this test. (1)

Given that the actual value of \(\mu\) is 14.9

(b)

(i) calculate the minimum value of \(n\) such that the probability of a Type II error is less than 0.05
Show your working clearly. (6)

(ii) Farhan uses the same sample size, \(n\), but now carries out the test at a 5% level of significance. Without doing any further calculations, state how this would affect the probability of a Type II error. (1)

Scheme	Marks	AO
Size of the test = 0.01	B1	1.2
	(1)

Mark scheme (b)

Scheme	Marks	AO
(i) Let CR be \(\overline{L} \lt k\)
\(\dfrac{k - 15}{\dfrac{0.2}{\sqrt{n}}} = -2.3263\)	M1	3.4
\(k = 15 - \dfrac{0.46526}{\sqrt{n}}\)	A1	1.1b
\(\dfrac{\text{“}15 - \dfrac{0.46526}{\sqrt{n}}\text{”} - 14.9}{\dfrac{0.2}{\sqrt{n}}} \gt 1.6449\)	M1d A1ft	3.4 1.1b
\(\dfrac{0.79424}{\sqrt{n}} \lt 0.1 \qquad \sqrt{n} \gt 7.9424\) oe	M1d	1.1b
\(n = 64\)	A1cso	2.1
	(6)
(ii) The probability of a Type II error would decrease.	B1	2.2a
	(1)
	(8 marks)

Notes

(i) M1: Finding the CR using the Normal distribution must have \(1.5 \lt |z| \lt 3.5\)

A1: A correct equation in the form \(k = \ldots\) and for use of awrt 2.326 (implied by awrt 0.46526 or awrt 0.46527)

M1d: Dependent on previous M being awarded. Standardising using their \(k\) and equating to a \(z\) value \(1.5 \lt |z| \lt 3\) to form an equation to able \(n\) to be found. May use = rather than >

A1ft: Ft their \(k\) for a correct equation with awrt 1.645

M1d: Dependent on previous M being awarded. Isolating \(\sqrt{n}\) or squaring both sides leading to a value for \(n\). Condone \(n = 7.9424\)

A1cso: 64 with correct working

(ii) B1: Suitable comment

ALT (b)(i)

Scheme	Marks	AO
\(\dfrac{k - 14.9}{\dfrac{0.2}{\sqrt{n}}} = 1.6449\)	M1	3.4
\(k = 14.9 + \dfrac{0.32898}{\sqrt{n}}\)	A1	1.1b
\(\dfrac{\text{“}14.9 + \dfrac{0.32898}{\sqrt{n}}\text{”} - 15}{\dfrac{0.2}{\sqrt{n}}} \gt -2.3263\)	M1d A1ft	3.4 1.1b
\(\dfrac{0.79424}{\sqrt{n}} \lt 0.1 \qquad \sqrt{n} \gt 7.9424\) oe	M1d	1.1b
\(n = 64\)	A1cso	2.1
	(6)