5. A midwife records the weights, in kg , of a sample of 50 babies born at a hospital. Her results are given in the table below.
| Weight ( \(\boldsymbol { w } \mathbf { ~ k g }\) ) | Frequency (f) | Weight midpoint (x) |
| \(0 \leqslant w < 2\) | 1 | 1 |
| \(2 \leqslant w < 3\) | 8 | 2.5 |
| \(3 \leqslant w < 3.5\) | 17 | 3.25 |
| \(3.5 \leqslant w < 4\) | 17 | 3.75 |
| \(4 \leqslant w < 5\) | 7 | 4.5 |
[You may use \(\sum \mathrm { f } x ^ { 2 } = 611.375\) ]
A histogram has been drawn to represent these data.
The bar representing the weight \(2 \leqslant w < 3\) has a width of 1 cm and a height of 4 cm .
- Calculate the width and height of the bar representing a weight of \(3 \leqslant w < 3.5\)
- Use linear interpolation to estimate the median weight of these babies.
- Show that an estimate of the mean weight of these babies is 3.43 kg .
- Find an estimate of the standard deviation of the weights of these babies.
Shyam decides to model the weights of babies born at the hospital, by the random variable \(W\), where \(W \sim \mathrm {~N} \left( 3.43,0.65 ^ { 2 } \right)\)
- Find \(\mathrm { P } ( W < 3 )\)
- With reference to your answers to (b), (c)(i) and (d) comment on Shyam's decision.
A newborn baby weighing 3.43 kg is born at the hospital.
- Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the
- mean,
- standard deviation.