June 2023 Paper 2 Q13
13 A large supermarket chain advertises that the mean mass of apples of a certain variety on sale in their stores is 0.14 kg.
Following a poor growing season, the head of quality control believes that the mean mass of these apples is less than 0.14 kg and she decides to carry out a hypothesis test at the 5% level of significance.
She collects a random sample of this variety of apple from the supermarket chain and records the mass, in kg, of each apple. She uses software to analyse the data. The results are summarised in the output below.
| n | 80 |
|---|---|
| Mean | 0.1316 |
| \(\sigma\) | 0.0198 |
| s | 0.0199 |
| \(\Sigma x\) | 10.525 |
| \(\Sigma x^2\) | 1.4161 |
| Min | 0.1 |
| Q1 | 0.12 |
| Median | 0.132 |
| Q3 | 0.1435 |
| Max | 0.19 |
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H_0}: \mu = 0.14\) \(\mathrm{H_1}: \mu < 0.14\) | B1 | 1.1 |
| their \(\mu\) is the population mean mass of this variety of apple | B1 | 2.5 |
| [2] |
Notes
B1: allow any other symbol except \(\bar{x}\) or \(\bar{X}\), as long as it is correctly defined;
allow hypotheses stated in words
B1: allow weight;
correct definition of \(\mu\) may be embedded in hypotheses written out as a sentence;
do not allow \(\bar{x}\) or \(\bar{X}\)
| Scheme | Marks | AO |
|---|---|---|
| \(\left[\bar{X} \sim\right] \mathrm{N}\left(0.14,\ \dfrac{0.0199^2}{80}\right)\) | B1 B1 | 3.3 2.2a |
| [2] |
Notes
B1: Normal distribution with correct mean or variance
allow variance = awrt \(4.95 \times 10^{-6}\) or awrt \(0.00222^2\)
B1: all correct, but allow full credit if no symbol used;
allow symbol other than \(\bar{X}\) if correctly defined as sample mean, but do not allow \(\mu\)
| Scheme | Marks | AO |
|---|---|---|
| awrt 0.136 seen BC | B1 | 1.1 |
| \(\bar{X} < 0.136\) only or \(\bar{X} \leqslant 0.136\) only | B1 | 3.4 |
| [2] |
Notes
B1: FT other correctly defined symbol
| Scheme | Marks | AO |
|---|---|---|
| \(0.1316 < 0.136\) or 0.1316 is in the critical region (must be correct critical region) oe or \(p =\) awrt \(0.00008 < 0.05\) oe NB 0.0000799 or \(z =\) awrt \(-3.78 < -1.645\) oe | M1 | 3.4 |
| reject \(\mathrm{H_0}\) | A1 | 1.1 |
| there is sufficient evidence at the 5% level to suggest that the mean mass of the apples is less than 0.14 kg | A1 | 2.2b |
| [3] |
Notes
M1: condone \(p =\) awrt \(0.00007 < 0.05\) oe NB 0.0000740
or \(z =\) awrt \(-3.79 < -1.645\) oe from use of \(\bar{X} \sim \mathrm{N}\left(0.14,\ \frac{0.0198^2}{80}\right)\)
A1: allow accept \(\mathrm{H_1}\) or result is significant
A1: allow weight; do not allow eg conclude / prove / indicate or other assertive statement instead of suggest