## (i)
**Answer/Working:** | B1 for cumulative frequencies; G1 for scales; G1 for labels; G1 for points (Provided plotted at correct UCB positions); G1 for joining points | 5

**Guidance:** May be implied from graph. Condone omission of 0 at this stage. Linear horizontal scale. Linear vertical scale: 0 to 50 (no inequality scales - Not even <9.1, <9.3, <9.5 ...). Heating quality or $x$ and Cumulative frequency or just CF or similar but not just frequency or fd nor cumulative fd. Plotted as (UCB, their cf). Ignore (9.1,0) at this stage. No midpoint or LCB plots. Plotted within ½ small square. For joining all of 'their points' (line or smooth curve) AND now including (9.1,0) dep on previous G1. Mid point or LCB plots may score first three marks. Can get up to 3/5 for cum freq bars. Allow full credit if axes reversed correctly

Lines of best fit could attract max 4 out of 5.

## (ii)
**Answer/Working:** Median = 9.67

**Marks:** B1 FT; Allow answers between 9.66 and 9.68 without checking curve. Otherwise check curve. | 3

**Guidance:** Based on 25th to 26th value on a cumulative frequency graph. If their mid-point plot (not LCB's) approx 9.57 for m.p. plot. Allow 9.56 to 9.58 without checking. B0 for interpolation

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# Question 8 continued

**Answer/Working:** $Q1 = 9.51$, $Q3 = 9.83$. Inter-quartile range = $9.83 - 9.51 = 0.32$

**Marks:** B1 FT for Q3 or Q1; B1 FT for IQR providing both Q1 and Q3 are correct. Allow answers between 9.50 and 9.52 and between 9.82 and 9.84 without checking curve. Otherwise check curve. | 3

**Guidance:** Based on 12th to 13th and 37th to 38th values on a cumulative frequency graph. If their mid-point plot (not LCB's) approx Q1 = 9.42; Q3 = 9.73. Allow 9.41 to 9.43 and 9.72 to 9.74 without checking. B0 for interpolation. Allow correct IQR from graph if quartiles not stated. Lines of best fit: B0 B0 B0 here.

## (iii)
**Answer/Working:** Lower limit $9.51 - 1.5 \times 0.32 = 9.03$. Upper limit $9.83 + 1.5 \times 0.32 = 10.31$. Thus there are no outliers in the sample.

**Marks:** B1 FT their $Q_1$, IQR; B1 FT their $Q_3$, IQR; E1; NB E mark dep on both B marks | 3

**Guidance:** Any use of median ± 1.5 IQR scores B0 B0 E0. If FT leads to limits above 9.1 or below 10.1 then E0. No marks for ± 2 or 3 IQR. In this part FT their values from (ii) if sensibly obtained (eg from LCB plot) or lines of best fit, but not from location ie 12.5, 37.5 or cumulative fx's or similar. For use of mean ± 2s, Mean = 9.652, s = 0.235, Limits 9.182, 10.122 gets M1 for correct lower limit, M1 for correct upper limit, zero otherwise, but E0 since there could be outliers using this definition

## (iv)
**Answer/Working:** (A) $P(\text{All 3 more than 9.5}) = \frac{38}{50} \times \frac{37}{49} \times \frac{36}{48} = 0.4304$ (=$50616/117600 = 2109/4900$)

**Marks:** M1 for 38/50 × (triple product); M1 for product of remaining fractions; A1 CAO | 3

**Guidance:** (38/50)³ which gives answer 0.4389 scores M1M0A0 so watch for this. M0M0A0 for binomial probability including 0.76¹⁰⁰ but ³$C_0×0.24^0×0.76^3$ still scores M1 ($k/50$)³ for values of $k$ other than 38 scores M0M0A0. M1M0A0 for binomial probability including 0.76¹⁰⁰ but ³$C_0×0.24^0×0.76^3$ still scores M1 ($k/50$)³ for values of $k$ other than 38 scores M0M0A0. $\frac{k}{50} \cdot \frac{(k-1)}{49} \cdot \frac{(k-2)}{48}$ for values of $k$ other than 38 scores M1M0A0

**Answer/Working:** (B) $P(\text{At least 2 more than 9.5}) = 3 \times \frac{38}{50} \times \frac{37}{49} \times \frac{12}{48} + 0.4304 = 3 \times 0.1435 + 0.4304 = 0.4304 + 0.4304 = 0.861$ (=$101232/117600 = 4218/4900 = 2109/2450$)

**Marks:** M1 for product of 3 correct fractions seen; M1 for 3 × a sensible triple or sum of 3 sensible triples; M1 indep for + 0.4304; FT (providing it is a probability); A1 CAO | 4

**Guidance:** Allow unsimplified fraction as final answer 50616/117600. Or (38/12)/50=0.4304 gets first two M1M1's. SC1 for $3 \times \frac{38}{50} \times \frac{38}{50} \times \frac{12}{50}$ or other sensible triple and SC2 if this + their 0.4304 (= 0.8549). Allow 0.86 or 2109/2450 or 4218/4900, but only M3A0 for other unsimplified fractions. Use of 1 – method 'with replacement' SC1 for $3 \times \frac{12}{50} \times \frac{12}{50} \times \frac{38}{50}$ SC2 for whole of $1 - 3 \times \frac{12}{50} \times \frac{12}{50} \times \frac{38}{50} - \frac{12}{50} \times \frac{12}{50} \times \frac{12}{50}$ (= 1 – (0.1313 + 0.0138) = 1 – 0.1451 = 0.8549)

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# NOTE RE OVER-SPECIFICATION OF ANSWERS

If answers are grossly over-specified (see instruction 8), deduct the final answer mark in every case. Probabilities should also be rounded to a sensible degree of accuracy. In general final non probability answers should not be given to more than 4 significant figures. Allow probabilities given to 5 sig fig. In general accept answers which are correct to 3 significant figures when given to 4 or 5 significant figures.