S1 January 2010 Q3
3. The birth weights, in kg, of 1500 babies are summarised in the table below.
| Weight (kg) | Midpoint, \(x\) kg | Frequency, f |
|---|---|---|
| 0.0 – 1.0 | 0.50 | 1 |
| 1.0 – 2.0 | 1.50 | 6 |
| 2.0 – 2.5 | 2.25 | 60 |
| 2.5 – 3.0 | 280 | |
| 3.0 – 3.5 | 3.25 | 820 |
| 3.5 – 4.0 | 3.75 | 320 |
| 4.0 – 5.0 | 4.50 | 10 |
| 5.0 – 6.0 | 3 |
[You may use \(\sum \mathrm{f}x = 4841\) and \(\sum \mathrm{f}x^2 = 15\,889.5\)]
| Scheme | Marks |
|---|---|
| 2.75 or \(2\tfrac{3}{4}\), 5.5 or 5.50 or \(5\tfrac{1}{2}\) | B1 B1 |
| (2) |
| Scheme | Marks |
|---|---|
| Mean birth weight \(= \dfrac{4841}{1500} = 3.227\dot{3}\) awrt 3.23 | M1 A1 |
| (2) |
Notes
M1 for a correct expression for mean. Answer only scores both.
| Scheme | Marks |
|---|---|
| Standard deviation \(= \sqrt{\dfrac{15889.5}{1500} - \left(\dfrac{4841}{1500}\right)^2} = 0.421093\ldots\) or \(s = 0.4212337\ldots\) | M1 A1ft A1 |
| (3) |
Notes
M1 for a correct expression (ft their mean) for sd or variance. Condone mis-labelling eg sd=… with no square root or no labelling
1st A1ft for a correct expression (ft their mean) including square root and no mis-labelling
Allow 1st A1 for \(\sigma^2 = 0.177\ldots \to \sigma = 0.42\ldots\)
2nd A1 for awrt 0.421. Answer only scores 3/3
| Scheme | Marks |
|---|---|
| \(Q_2 = 3.00 + \dfrac{403}{820} \times 0.5 = 3.2457\ldots\) (allow 403.5….. \(\to\) 3.25) | M1 A1 |
| (2) |
Notes
M1 for a correct expression (allow 403.5 i.e. use of \(n + 1\)) but must have 3.00, 820 and 0.5
A1 for awrt 3.25 provided M1 is scored.
NB 3.25 with no working scores 0/2 as some candidates think mode is 3.25.
| Scheme | Marks |
|---|---|
| Mean(3.23)<Median(3.25) (or very close) | B1ft |
| Negative Skew (or symmetrical) | dB1ft |
| (2) | |
| (11 marks) |
Notes
1st B1ft for a comparison of their mean and median (may be in a formula but if ±(mean - median) is calculated that’s OK. We are not checking the value but the sign must be consistent.)
Also allow for use of quartiles provided correct values seen: \(Q_1 = 3.02, Q_3 = 3.47\)
[They should get \((0.22 =)\, Q_3 - Q_2 \lt Q_2 - Q_1\, (= 0.23)\) and say (slight) negative skew or symmetric]
2nd dB1ft for a compatible comment based on their comparison. Dependent upon a suitable, correct comparison. Mention of “correlation” rather than “skewness” loses this mark.