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 compute the median, we first determine the cumulative frequency:

• Up to 2 kg: 1
• Up to 3 kg: 9
• Up to 3.5 kg: 26

The median is the value at the rac{50}{2} = 25th position, which falls into the interval of 3 ≤ w < 3.5:

Using linear interpolation:

Let L = 3, C.F = 9, F = 17, h = 0.5

The median weight is estimated as:

ext{Median} = L + rac{(n/2 - C.F)}{f} imes h

Calculating:

ext{Median} = 3 + rac{(25 - 9)}{17} imes 0.5 \\ ext{Median} acksimeq 3.47

To find the mean weight, apply:

ext{Mean} = rac{ ext{Σ}(f imes x)}{N}

Calculating Σ(f × x):

$= (1 × 1) + (8 × 2.5) + (17 × 3.25) + (17 × 3.75) + (7 × 4.5) = 171.5$

Dividing by the total frequency (N = 50):

ext{Mean} = rac{171.5}{50} = 3.43 ext{ kg}

Calculate variance using:

ext{Variance} = rac{ ext{Σ}(f imes (x - ext{Mean})^2)}{N}

We compute Σ(f × (x - 3.43)²) to find the standard deviation, with the computed variance:

$ext{Standard Deviation} = ext{√Variance}$

After calculations, we estimate:

ext{Standard Deviation} acksimeq 0.680

Using the normal distribution model for Shyam's decision:

Convert to the standard normal variable:

Z = rac{3 - 3.43}{0.65} acksimeq -0.662

Using Z-tables to find:

P(W < 3) acksimeq P(Z < -0.662) acksimeq 0.2546

Shyam's normal approximation suggests a mean of 3.43 kg, which aligns with the estimated mean from our calculations. However, variability (as indicated by standard deviation) may not accurately represent the sample data. A potentially skewed distribution would imply less reliability in Shyam's statistical modeling.

Adding a newborn baby weighing 3.43 kg will not affect the mean significantly, as it equals the current mean. Thus, the mean remains unchanged.

The addition of a newborn baby weighing 3.43 kg will decrease the standard deviation slightly, as the new data point coincides with the mean, resulting in a decreased spread of weights.