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 estimate the median weight, first, determine the cumulative frequency. The total number of babies is 50, so the median is the 25th baby. From the cumulative frequencies:

• 1 baby (0 ≤ w < 2)
• 9 babies (0 ≤ w < 3)
• 26 babies (0 ≤ w < 3.5)

The median falls in the interval of 3 ≤ w < 3.5. To find its exact position:

Using linear interpolation:

$\text{Median} = 3 + \left( \frac{25 - 26}{17} \right) \times (3.5 - 3) = 3 + \left( \frac{-1}{17} \right) \times 0.5$

Calculating this gives:

$\text{Median} \approx 3 - 0.0294 \approx 3.0006 \text{ kg}$

To calculate the mean weight, use the formula:

$\text{Mean} = \frac{\sum (f \cdot x)}{N}$

Where:

• f is the frequency of each weight interval,
• x is the weight midpoint,
• N is the total frequency (50).

Calculating the contributions:

$\sum (f \cdot x) = (1 \cdot 1) + (8 \cdot 2.5) + (17 \cdot 3.25) + (17 \cdot 3.75) + (7 \cdot 4.5)$

This equates to:

$= 1 + 20 + 55.25 + 63.75 + 31.5 = 171.5$

Thus:

$\text{Mean} = \frac{171.5}{50} = 3.43 \text{ kg}$

The standard deviation can be estimated using:

$SD = \sqrt{\frac{\sum f (x - \text{Mean})^2}{N}}$

Calculating:

1. Find $(x - \text{Mean})^2$ for each interval,
2. Multiply by frequency and sum.

After calculating, using the estimate of the mean:

1. Compute the sum of squares.
2. Finally divide by N and take the square root to find SD.

This gives:

$SD \approx 0.68 \text{ kg}$

To find the probability:

$P(W < 3) = P\left(Z < \frac{3 - 3.43}{0.65}\right)$

Calculating the Z-score:

$Z = \frac{-0.43}{0.65} \approx -0.6615$

Using Z-tables:

$P(Z < -0.6615) \approx 0.2546 \text{ or } 0.254 \text{ (in decimal form)}$

Referring to the median, mean, and standard deviation:

• The median estimated (3.0006 kg) is lower than the mean (3.43 kg), potentially indicating a right-skewed distribution.
• Shyam’s model assumes a normal distribution, which may not accurately reflect the data as it's skewed.

Thus, his use of a normal distribution could lead to inaccurate predictions.

The addition of a baby weighing 3.43 kg will not change the mean, as it equals the current mean of the sample. Thus, it remains 3.43 kg.

The standard deviation will decrease because adding a data point that is equal to the mean reduces the spread of the data. New data points that are closer to the mean typically decrease standard deviation.