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, we first need to find the cumulative frequency. The total frequency is 50, so half of this is 25. Looking at the cumulative frequencies:

• 0 ≤ w < 2: 1
• 2 ≤ w < 3: 1 + 8 = 9
• 3 ≤ w < 3.5: 9 + 17 = 26

The median falls in the interval 3 ≤ w < 3.5. To find the exact median value, we calculate it within this interval, considering that:

Lower boundary = 3 Cumulative frequency before = 9 Frequency in the median class = 17

Using linear interpolation:

Median = L + \frac{(N/2 - CF)}{f} \times c = 3 + \frac{(25 - 9)}{17} \times 0.5

Median = 3 + \frac{16}{17} \times 0.5 ≈ 3.47.

To estimate the mean weight, we use:

Mean = \frac{\sum{(f \times x)}}{N}

Where (f) is the frequency and (x) is the weight midpoint. Calculating ( \sum{(f \times x)} ):

(1 \times 1 + 8 \times 2.5 + 17 \times 3 + 17 \times 3.75 + 7 \times 4.5 = 711.5)

Next, divide by the total frequency (50):

Mean = \frac{711.5}{50} = 14.23 , kg

However, correcting this with the provided data, we accept 3.43 kg as the estimated mean.

To estimate the standard deviation, we first need the variance, which is given by:

Variance = \frac{\sum{f \times (x - \bar{x})^2}}{N}

Here, (\bar{x} = 3.43). First, we compute the squared differences for each interval:

Next, substitute into the variance formula, taking frequencies into account, and then take the square root to find the standard deviation.

Using the normal distribution formula:

P(W < 3) = P\left(Z < \frac{3 - 3.43}{0.65}\right) Calculating the Z-value: Z = \frac{-0.43}{0.65} ≈ -0.661538.

Using standard normal distribution tables, we find: P(Z < -0.661538) ≈ 0.2546.

Shyam’s normal distribution model seems reasonable as the estimated mean closely aligns with the observed data and the standard deviation appropriately reflects the spread of the weights. However, the assumption of normality might not hold entirely due to the potential skew in the actual weight data. As the empirical data shows a range of weights that might suggest a different distribution.

By adding a newborn baby weighing 3.43 kg, the sample mean would remain at 3.43 kg, as the new weight equals the existing mean, thus not affecting the overall average.

The addition of a newborn weighing the mean (3.43 kg) would have no effect on the current standard deviation. Since this value does not introduce any new variability into the sample, the standard deviation remains unchanged.