## Question 4:

### Part (i)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Frequency densities: $30 \le w < 50$: fd $= 0.55$; $50 \le w < 60$: fd $= 1$; $60 \le w < 70$: fd $= 1.8$; $70 \le w < 80$: fd $= 1.4$; $80 \le w < 90$: fd $= 0.7$ | M1 | For fd's — at least 3 correct. Accept any suitable unit for fd such as freq per 10g. M1 can also be gained from freq per: 5.5, 10, 18, 14, 7 (at least 3 correct) or similar. If fd not explicitly given, M1 and A1 can be gained from all heights correct (within half a square) on histogram |
| All fd's correct | A1 | |
| Linear scales on both axes and labels | G1 | Linear scale and label on vertical axis in relation to first M1 mark, i.e. fd or frequency density, or if relevant freq/10, etc (NOT e.g. fd/10). Allow scale given as fd$\times$10 or similar. Accept f/w or f/cw. Ignore horizontal label |
| Width of bars correct (no gaps, drawn at 30, 50, etc.) | G1 | Must be drawn at 30, 50, 60 etc. NOT 29.5 or 30.5 etc. No gaps allowed. No inequality labels on their own such as $30 \le W < 50$ etc but allow if 30, 50, 60 etc occur at correct boundary position |
| Height of bars correct | G1 | Height of bars — must be linear vertical scale. FT of heights dep on at least 3 heights correct and all must agree with their fds. If fds not given and at least 3 heights correct then max M1A0G1G1G0. Allow restart with correct heights if given fd wrong (for last three marks only) |

### Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Mean} = \frac{(40\times11)+(55\times10)+(65\times18)+(75\times14)+(85\times7)}{60} = \frac{3805}{60}$ | M1 | For midpoints (at least 3 correct): products are 440, 550, 1170, 1050, 595. No marks for mean or sd unless using midpoints |
| $= 63.4$ (or 63.42) | A1 | CAO (exact answer 63.41666…). Answer must NOT be left as improper fraction as this is an estimate |
| $\sum x^2 f = (40^2\times11)+(55^2\times10)+(65^2\times18)+(75^2\times14)+(85^2\times7) = 253225$ | | |
| $S_{xx} = 253225 - \frac{3805^2}{60} = 11924.6$ | M1 | For attempt at $S_{xx}$. Should include sum of at least 3 correct multiples $fx^2 - \frac{\sum x^2}{n}$. Allow M1 for anything which rounds to 11900. Use of mean 63.4 leading to answer of 14.29199… with $S_{xx} = 12051.4$ gets full credit |
| $s = \sqrt{\frac{11924.6}{59}} = \sqrt{202.11} = 14.2$ | A1 | At least 1dp required. 63.42 leads to 14.2014… Do not FT their incorrect mean (exact answer 14.2166…). Allow SC1 for RMSD 14.1 (14.0976…) from calculator. If using $(x-\bar{x})^2$ method, B2 if 14.2 or better (14.3 if use of 63.4), otherwise B0 |

## Question 4(iii):

$\bar{x} - 2s = 63.4 - (2 \times 14.2) = 35$ | M1 | For either; No marks in (iii) unless using $\bar{x} + 2s$ or $x - 2s$; Only follow through numerical values, not variables such as $s$; if candidate does not find $s$ but writes 'limit is $63.4 + 2 \times$ standard deviation', do NOT award M1; Do not penalise for over-specification

$\bar{x} + 2s = 63.4 + (2 \times 14.2) = 91.8$ | A1 | For both (FT); Must have correct limits to get this mark

So there are probably some outliers at the lower end, but none at the upper end | E1 | Must include an element of doubt and must mention both ends

**Total: [3]**

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## Question 4(iv):

$\text{Mean} = \dfrac{3624.5}{50} = 72.5\text{g}$ (or exact answer 72.49g) | B1 | CAO Ignore units

$S_{xx} = 265416 - \dfrac{3624.5^2}{50} = 2676$ | M1 | For $S_{xx}$; M1 for $265416 - 50 \times \text{their mean}^2$; BUT NOTE M0 if their $S_{xx} < 0$; For $s^2$ of 54.6 (or better) allow M1A0 with or without working

$s = \sqrt{\dfrac{2676}{49}} = \sqrt{54.61} = 7.39\text{g}$ | A1 | CAO ignore units; Allow 7.4 but NOT 7.3 (unless RMSD with working); For RMSD of 7.3 (or better) allow M1A0 provided working seen; For RMSD$^2$ of 53.5 (or better) allow M1A0 provided working seen

**Total: [3]**

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## Question 4(v):

Variety A have lower average than Variety B | E1 | FT their means; Do not condone 'lower central tendency' or 'lower mean'; Allow 'on the whole' or similar in place of 'average'

Variety A have higher variation than Variety B | E1 | FT their sd; Allow 'more spread' or similar but not 'higher range' or 'higher variance'; Condone 'less consistent'

**Total: [2]**