## (i)
**Answer/Working:** Mean = $\frac{1 \times 10 + 2 \times 40 + 3 \times 15 + 4 \times 5}{70} = \frac{155}{70} = 2.214$

$S_{xx} = 1^2 \times 10 + 2^2 \times 40 + 3^2 \times 15 + 4^2 \times 5 - \frac{155^2}{70} = 385 - 343.21 = 41.79$

$s = \sqrt{\frac{41.79}{69}} = 0.778$

**Marks:** M1, A1 CAO | 5

**Guidance:** For M1 allow sight of at least 3 double pairs seen from $1 \times 10 + 2 \times 40 + 3 \times 15 + 4 \times 5$ with divisor 70. Allow answer of 155/70 = 2.2 or 2.21 or 31/14 oe. For 155/70 = eg 2.3 , allow A1 isw. M1 for $\Sigma fx^2$ s.o.i. M1 for attempt at $S_{xx}$, Dep on first M1. M1 for $l^2 \times 10 + 2^2 \times 40 + 3^2 \times 15 + 4^2 \times 5$ with at least three correct terms. Using exact mean leads to $S_{xx} = 41.79$, $s=0.778$. Using mean 2.214 leads to $S_{xx} = 41.87$, $s=0.779$. Using mean 2.21 leads to $S_{xx} = 43.11$ and $s = 0.790$. Using mean 2.2 leads to $S_{xx} = 46.2$ and $s = 0.818$. Using mean 2 leads to $S_{xx} = 105$ and $s = 1.233$. All the above get M1M1A1 except the last one which gets M1M1A0. RMSD(divisor $n$ rather than $n-1$) = $\sqrt{(41.79/70)} = 0.772$ gets M1M1A0. Alternative method, award M1for at least 3 terms of and second M1 for all 4 terms of $(1-2.214)^2 \times 10 + (2-2.214)^2 \times 40 + (3-2.214)^2 \times 15 + (4-2.214)^2 \times 5(= 41.79)$. NB Allow full credit for correct answers without working (calculator used)

## (ii)
**Answer/Working:** Mean would decrease, Standard deviation would increase

**Marks:** B1, B1 | 2

**Guidance:** Do not accept increase/decrease seen on their own – must be linked to mean and SD. Allow eg 'It would skew the mean towards zero' and eg ' It would stretch the SD'. SC1 for justified argument that standard deviation might either increase or decrease according to number with no eggs ($n \leq 496$ increase, $n \geq 497$ decrease)

---