June 2025 Paper 2 Q17
17 In this question you must show detailed reasoning
Jars of honey contain a nominal mass of 340 grams of honey.
The jars are filled by an automated process so that the actual mass of honey in a jar, \(X\) grams, is Normally distributed with mean 350 grams. This is to minimize the chance of a jar which contains less than 340 grams of honey being sold.
During a quality control check, a random sample of 50 jars of honey are checked.
Summary statistics for the check are as follows.
\(\sum x = 17\,367 \qquad \sum x^2 = 6\,036\,597 \qquad n = 50\)
Carry out a hypothesis test at the 1% level to see if there is any evidence to suggest that the mean mass of honey in the jars is less than 350 grams. [9]
| Scheme | Marks | AO |
|---|---|---|
| DR \([\bar{x} =]\ 347.34\) seen | B1 | 1.1 |
| \(\left(s^2 = \dfrac{6036597 - 50\times 347.34^2}{49}\right) = 88.637\) or \((s =)\ 9.4147\) seen BC | B1 | 1.1 |
| \(\mathrm{H_0}\): \(\mu = 350\) \(\mathrm{H_1}\): \(\mu \lt 350\) | B1 | 1.1 |
| \(\mu\) is the population mean mass of honey in a jar | B1 | 2.5 |
| \(\mathrm{N}\left(350, \dfrac{\text{awrt } 88.6}{50}\right)\) oe seen | B1 | 3.3 |
| \((\mathrm{P}(\bar{X} \lt 347.34) =)\) awrt 0.023 | M1* | 3.4 |
| their 0.02287 correctly compared with 0.01 | M1dep* | 3.4 |
| do not reject \(\mathrm{H_0}\) or accept \(\mathrm{H_0}\) or reject \(\mathrm{H_1}\) or not significant | A1FT | 1.1 |
| there is insufficient evidence at the 1% level to suggest that the mean mass of honey in a jar is less than 350 grams | A1 | 2.2b |
| [9] |
Notes
DR: This question included the instruction: In this question you must show detailed reasoning.
B1: BC allow 347.3 or 347; allow eg mean or \(\bar{X}\), but no other symbol if seen
B1: NB 88.6371428571; may see 9.4147(3010007); accept awrt 88.6 or awrt 9.41; do not allow 86.86 or 9.320; allow eg \(\sigma^2\) or VAR[\(X\)]
B1: allow use of other parameters (but not \(\bar{x}\) or \(\bar{X}\)) if defined as [population] mean mass;
allow if hypotheses stated in words: may earn B1B0 if “population” omitted
B1: condone weight instead of mass
B1: FT their \(s^2\) correct to 3 sf or better;
may see N(350, 1.77(274285741)) or \(\mathrm{N}\left(350, \dfrac{9.415^2}{50}\right)\) or \(\mathrm{N}(350, 1.331^2)\)
M1*: allow M1 for correct probability found from \(\mathrm{N}\left(350, \dfrac{\text{their awrt } 88.6}{50}\right)\);
must see their distribution if value incorrect;
condone slip in notation eg use of \(\mathrm{P}(X \lt 347.34)\); allow for use of 347 or 347.3 instead of 347.34
M1dep*: allow eg 2.28% \(\gt\) 1%
A1FT: A0 if their 0.0229 \(\lt\) 0.01
A1: do not allow eg conclude / prove / indicate or other assertive statement instead of suggest; this mark is dependent on the award of all preceding marks apart from the fourth and fifth B marks ; condone weight instead of mass; must come from correct probability: \((\mathrm{P}(\bar{X} \lt 347.34) =)\) awrt 0.023
Alternatively, for the last 4 marks
| Scheme | Marks |
|---|---|
| critical region is \(\bar{X} \lt 346.9(0)\) or critical value is \(\bar{X} = 346.9(0)\) to 3 or more sf | M1* |
| 347.34 correctly compared with their 346.90 | M1dep* |
| do not reject \(\mathrm{H_0}\) or accept \(\mathrm{H_0}\) or reject \(\mathrm{H_1}\) or not significant | A1FT |
| there is insufficient evidence at the 1% level to suggest that the mean mass of honey in a jar is less than 350 grams | A1 |
M1*: allow M1 for critical region found from \(\mathrm{N}\left(350, \dfrac{\text{their awrt } 88.6}{50}\right)\); must see their distribution if value incorrect; condone slip in notation eg use of \(X\)
M1dep*: allow 347 or 347.3 compared with 346.9(0)
A1FT: A0 if their 347.34 \(\lt\) their 346.9(0)
A1: do not allow eg conclude / prove / indicate or other assertive statement instead of suggest; this mark is dependent on the award of all preceding marks apart from the fourth and fifth B marks; must have correct critical region
Alternatively, for the last 4 marks
| Scheme | Marks |
|---|---|
| \(z\left[= \dfrac{347.34 - 350}{\frac{9.414}{\sqrt{50}}}\right] = \text{awrt } -1.998\) | M1* |
| their \(-1.998 \gt -2.33\) | M1dep* |
| do not reject \(\mathrm{H_0}\) or accept \(\mathrm{H_0}\) or reject \(\mathrm{H_1}\) or not significant | A1FT |
| there is insufficient evidence at the 1% level to suggest that the mean mass of honey in a jar is less than 350 grams | A1 |
M1*: allow M1 for \(z\) found from \(\mathrm{N}\left(350, \dfrac{\text{their awrt } 88.6}{50}\right)\); must see their distribution if value incorrect; allow for use of 347 or 347.3 instead of 347.34
NB may see awrt \(-2.03\) or awrt \(-2.25\) from 347.3 or 347
M1dep*: their \(z\) correctly compared with \(-2.326347877\ldots\) to 2 or more dp
A1FT: A0 if their \(-1.998 \lt -2.33\)
A1: do not allow eg conclude / prove / indicate or other assertive statement instead of suggest; this mark is dependent on the award of all preceding marks apart from the fourth and fifth B marks; \(z\) must be correct