A midwife records the weights, in kg, of a sample of 50 babies born at a hospital - Edexcel - A-Level Maths Statistics - Question 5 - 2016 - Paper 1

To find the median, calculate $n/2 = 50/2 = 25$. The cumulative frequencies indicate that the 25th weight falls within the range of $3 ≤ w < 3.5$. Thus, using weights 3 and 3.5, we apply linear interpolation:

Median weight = $3 + \left(\frac{25 - 24}{17}\right) \times (3.5 - 3) = 3 + \frac{1}{17} \times 0.5 = 3.0294$ kg.

To compute the mean weight, we use the formula:

Mean = $\frac{\sum (f \cdot x)}{n}$,

Where $f$ is frequency and $x$ is the weight midpoint. Substituting into the formula gives:

Mean = $\frac{1 \cdot 1 + 8 \cdot 2.5 + 17 \cdot 3 + 17 \cdot 3.75}{50} = \frac{171.5}{50} = 3.43$ kg.

To calculate standard deviation, use:

$\sigma = \sqrt{\frac{\sum (f \cdot (x - \bar{x})^2)}{n}}$,

Where $\bar{x}=3.43$. Calculating:

Variance $= \frac{1 \cdot (1 - 3.43)^2 + 8 \cdot (2.5 - 3.43)^2 + 17 \cdot (3 - 3.43)^2 + 17 \cdot (3.75 - 3.43)^2}{50-1} = 0.85$,

Standard deviation $= \sqrt{0.85} \approx 0.9219$.

Using the normal distribution, we need to standardize:

$Z = \frac{X - \mu}{\sigma} = \frac{3 - 3.43}{0.65} = -0.6615$. From the Z-table, we find:

P(W < 3) = P(Z < -0.6615) = 0.2546$.

Shyam's model anticipates that the weights are normally distributed. Given that the mean weight is close to 3.43 kg and the standard deviation is relatively low (0.65 kg), the normal model is a reasonable approximation. However, the assumption of normality should be checked considering the distributions in the original frequency table.

This implies the new baby weighs exactly at the mean. Consequently, the mean weight would remain unchanged while the variance is potentially affected. An additional newborn would contribute to greater data stability, thus standard deviation might decrease or slightly increase, reflecting the concentration around the mean.