## Part (i)

**Answer:** $4 + \frac{1}{2} \text{ of } 18 = 4 + 9 = 13$

| **Marks** | **Guidance** |
|-----------|--------------|
| M1 | For $\frac{1}{2}$ of 18 |
| A1 | cao |
| **[2]** | 13/100 gets M1A0 |

## Part (ii)

**Answer:** (Median) = $50.5^{\text{th}}$ value

$\text{Est} = 140 + \left(\frac{25.5}{29}\right) \times 5$ or $140 + \left(\frac{50.5 - 25}{54 - 25}\right) \times 5 = 144.4$

| **Marks** | **Guidance** |
|-----------|--------------|
| M1 | For $50.5$ seen. SC2 for use of $50^{\text{th}}$ value leading to Est $= 140 + (25/29 \times 5) = 144.3$ (SC1 if over-specified). Or Est $= 145 - \left(\frac{3.5}{29}\right) \times 5 = 144.4$ |
| M1 | For attempt to find this value |
| A1 | |
| **[3]** | NB no marks for mean $= 144.35$. NB Watch for over-specification |

## Part (iii)

**Answer:** Frequency densities: 1.67, 5.80, 4.80, 1.80, 0.40

| **Marks** | **Guidance** |
|-----------|--------------|
| M1 | For fd's - at least 3 correct. Accept any suitable unit for fd such as freq per cm. M1 can be also be gained from freq per 10 – 16.7, 58, 48, 18, 4 (at least 3 correct) or freq per 5 – 8.35, 29, 24, 9, 2 for all correct. If fd not explicitly given, M1 A1 can be gained from all heights correct (within one square) on histogram (and M1A0 if at least 3 correct) |
| G1 | linear scales on both axes and label on vertical axis IN RELATION to first M1 mark ie fd or frequency density or if relevant freq/10, etc (NOT eg fd/10). However allow scale given as fdx10, or similar. Accept f/w or f/cw (freq/class width). Can also be gained from an accurate key. G0 if correct label but not fd's. Must be drawn at 125, 140 etc NOT 124.5 or 125.5 etc NO GAPS ALLOWED. Must have linear scale. No inequality labels on their own such as $125 \leq S < 140$, etc but allow if a clear horizontal linear scale is also given. Ignore horizontal label. |
| W1 | width of bars. |
| H1 | height of bars. Height of bars – must be linear vertical scale. FT of heights dep on at least 3 heights correct and all must agree with their |
| **[5]** | fds. If fds not given and at least 3 heights correct then max M1A0G1W1H0. Allow restart with correct heights if given fd wrong (for last three marks only) |

## Part (iv)

**Answer:** 0.6 × 15 = 9 girls. So 5 more girls

| **Marks** | **Guidance** |
|-----------|--------------|
| M1 | For 0.6 × 15 |
| A1 | For 9 girls |
| A1 | cao |
| **[3]** | Or 45 × 0.2 = 9 (number of squares and 0.2 per square) |

## Part (v)

**Answer:** Frequencies and midpoints for girls are:
| Height | 132.5 | 142.5 | 147.5 | 155 | 167.5 |
|--------|-------|-------|-------|-----|-------|
| Frequency | 18 | 23 | 31 | 19 | 9 |

So mean = $\frac{(132.5 \times 18) + (142.5 \times 23) + (147.5 \times 31) + (155 \times 19) + (167.5 \times 9)}{100}$

$= \frac{(2385) + (3277.5) + (4572.5) + (2945) + (1507.5)}{100} = 146.9$ (Exact answer 146.875)

| **Marks** | **Guidance** |
|-----------|--------------|
| B1 | For at least three frequencies correct |
| B1 | At least three midpoints correct. No further marks if not using midpoints |
| M1 | For attempt at $\sum xy$ |
| M1* | For division by 100 |
| A1 | cao |
| **[5]** | For sight of at least 3 xy pairs. Allow answer 146.9 or 147 but not 150. NB Accept answers seen without working (from calculator). Use of 'not quite right' midpoints such as 132.49 or 132.51 etc can get B1B0M1M1A0. NB Watch for over-specification |

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