June 2024 Paper 3 Q19

AQACurrent spec9 marksIncludes hypothesis testingBinomial Distribution Large Data Set

19 It is known that 80% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than 0.3 g/km.

Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than 0.3 g/km has changed.

Talat will carry out a hypothesis test at the 10% significance level on a random sample of 25 diesel cars registered in 2022.

(a)

(i) State suitable null and alternative hypotheses for Talat’s test. [1 mark]

(ii) Using a 10% level of significance, find the critical region for Talat’s test. [5 marks]

(iii) In his random sample, Talat finds 18 cars with CO emissions less than 0.3 g/km.

State Talat’s conclusion in context. [1 mark]

(b) Talat now wants to use his random sample of 25 diesel cars, registered in 2022, to investigate whether the proportion of diesel cars in England with CO emissions more than 0.5 g/km has changed from the proportion given by the Large Data Set.

Using your knowledge of the Large Data Set, give two reasons why it is not possible for Talat to do this. [2 marks]

Mark scheme (a)

Scheme	Marks	AO
(i) States \(\mathrm{H_0} : p = 0.8\) \(\mathrm{H_1} : p \neq 0.8\)	B1	2.5
	(1)
(ii) States or uses correct model PI by calculation of one of \(\mathrm{P}(X \leqslant x)\) where \(x\) = [1, 24] or \(\mathrm{P}(X \geqslant x)\) where \(x\) = [1, 25] or \(\mathrm{P}(X = x)\) where \(x\) = 0 or 25 or by critical region of \(x \leqslant 16\) or \(x \geqslant 24\)	B1	3.3
Obtains one of (List 1) [0.017, 0.0174] or [0.046, 0.047] or [0.109, 0.11] or [0.098, 0.0983] or [0.027, 0.0274] or [0.0037, 0.0038] or obtains one of (List 2) [0.982, 0.983] or [0.953, 0.954] or [0.890, 0.891] or [0.901, 0.902] or [0.97, 0.973] or [0.996, 0.9963] PI by critical region of \(x \leqslant 16\) or \(x \geqslant 24\) Ignore labels	M1	1.1a
Compares one probability from List 1 with 0.05 or compares one probability from List 2 with 0.95 PI by critical region of \(x \leqslant 16\) or \(x \geqslant 24\)	M1	1.1a
Obtains one of the critical regions of \(x \leqslant 16\) or \(x \geqslant 24\)	A1	1.1b
Obtains both critical regions \(x \leqslant 16\), \(x \geqslant 24\) Accept critical region of \(x \lt 17\) or \(0 \leqslant x \lt 17\), \(x \gt 23\) Condone use of any letter for \(x\) or stating \(\leqslant 16\) or \(\geqslant 24\) throughout	A1	3.2a
	(5)
(iii) Concludes correctly in context from a fully correct comparison Conclusion must not be definite, eg use of ‘suggest’, ‘support’ etc Follow through the correct comparison of 18 with their critical region from part 19(a)(ii) The comparison does not need to be seen	E1F	2.2b
	(1)

Typical solution

(i)

\[\mathrm{H_0} : p = 0.8\]\[\mathrm{H_1} : p \neq 0.8\]

(ii)

\[X \sim \mathrm{B}(25, 0.8)\]\[\mathrm{P}(X \leqslant 15) = 0.0173 \lt 0.05\]\[\mathrm{P}(X \leqslant 16) = 0.0468 \lt 0.05\]\[\mathrm{P}(X \leqslant 17) = 0.1091 \gt 0.05\]\[\mathrm{P}(X \geqslant 23) = 0.0982 \gt 0.05\]\[\mathrm{P}(X \geqslant 24) = 0.0274 \lt 0.05\]\[\mathrm{P}(X \geqslant 25) = 0.0038 \lt 0.05\]

Critical region is \(x \leqslant 16\), \(x \geqslant 24\)

(iii)

There is insufficient evidence to suggest that proportion of diesel cars registered in 2022 with CO emissions less than 0.3 g/km has changed

Mark scheme (b)

Scheme	Marks	AO
States one of the following reasons no diesel cars in the LDS have CO emissions more than 0.5 g/km Values of CO emissions for some diesel cars are missing from LDS LDS data is not representative due to only few regions in England or some makes of cars or only two years	E1	2.4
States a different reason no diesel cars in the LDS have CO emissions more than 0.5 g/km Values of CO emissions for some diesel cars are missing from LDS LDS data is not representative due to only few regions in England or some makes of cars or only two years	E1	2.4
	(2)
	(9 marks)

Typical solution

No diesel cars in the LDS have CO emissions more than 0.5 g/km

The LDS only uses data from cars in some regions in England but not all