June 2025 Paper 2 Q13
13 According to the Office for National Statistics, in 2022 the proportion of people in England and Wales aged 16 or over who were married or in a civil partnership was 0.494.
In 2024 a researcher collected a random sample of 78 people aged 16 or over.
The researcher decided to conduct a hypothesis test at the 1% level to see if there was any evidence to suggest that the proportion of people aged 16 or over who are married or in a civil partnership was lower in 2024.
The researcher found that 26 of the 78 people in the random sample were married or in a civil partnership.
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H_0}\ p = 0.494\) \(\mathrm{H_1}\ p \lt 0.494\) | B1 | 1.1 |
| \(p\) is the probability that a person (aged 16 or over from England or Wales selected at random) is married or in a civil partnership | B1 | 2.5 |
| [2] |
Notes
B1: allow equivalent in words; do not allow percentages
B1: or \(p\) is the proportion of people (aged 16 or over from England or Wales selected at random) who are married or in a civil partnership; must mention either married or in a civil partnership
B1B1 for other symbol instead of \(p\) used if correctly defined
| Scheme | Marks | AO |
|---|---|---|
| use of \(X \sim \mathrm{B}(78, 0.494)\) soi | M1 | 3.3 |
| \((\mathrm{P}(X \leqslant 27) =)\ 0.0059\ldots\) and \((\mathrm{P}(X \leqslant 28) =)\ 0.011\ldots\) seen | M1 | 3.4 |
| \([0 \leqslant]\ X \leqslant 27\) or \([0 \leqslant]\ X \lt 28\) oe | A1 | 1.1 |
| [3] |
Notes
M1: for calculation of a value for \(\mathrm{P}(X \leqslant k)\), ignore slip in notation
M0 for attempt at Normal approximation
M1: may see invBinom(0.01,78,0.494) = 28 for M1M1;
A1: allow critical region \(\lt 28\) oe; if M0M0 allow SC1 for correct critical region with no working or insufficient working
| Scheme | Marks | AO |
|---|---|---|
| 26 correctly compared with their 27 or their 28 | M1 | 3.4 |
| reject \(\mathrm{H_0}\) or accept \(\mathrm{H_1}\) or significant | A1 | 1.1 |
| there is sufficient evidence at the 1% level to suggest that the proportion of people aged 16 or over who are married or in a civil partnership is lower in 2024 than in 2022 | A1 | 2.2b |
| [3] |
Notes
M1: allow eg 26 is in critical region
A1: from correct critical region; condone from \(0 \lt X \leqslant 27\) oe
A1: dependent on award of previous two marks and correct critical region; allow percentage/probability but not “number”
do not allow eg conclude / prove / indicate or other assertive statement instead of suggest;
if M0 allow SC1 for \(\mathbf{P}(\boldsymbol{X} \leqslant \mathbf{26}) = \mathbf{0.003} \lt \mathbf{0.01}\), so reject \(\mathrm{H_0}\) and conclude there is sufficient evidence at the 1% level to suggest that the proportion of people aged 16 or over who are married or in a civil partnership is lower in 2024 than in 2022
| Scheme | Marks | AO |
|---|---|---|
| 0.01 oe | B1 | 1.2 |
| [1] |
Notes
B1: allow \([\mathrm{P}(X \leqslant 27) =]\ 0.005\,946\ldots\)