The birth weights, in kg, of 1500 babies are summarised in the table below - Edexcel - A-Level Maths Statistics - Question 3 - 2010 - Paper 1

To estimate the mean birth weight, we use the formula:

ext{Mean} = rac{ ext{Total} ext{ of } (fx)}{N}

Where

• \( ext{Total} ext{ of } (fx) = 4841 \)
• N = 1500 (total frequency)

Calculating:

ext{Mean} = rac{4841}{1500} ≈ 3.2274

Thus, the estimated mean birth weight is approximately 3.23 kg.

The standard deviation can be estimated using the formula:

ext{Standard Deviation} = \\sqrt{rac{ ext{Total}(fx^2)}{N} - ext{Mean}^2}

Here,

• Total $fx^2 = 15889.5$
• Mean from part (b) is approximately $3.2274$
• N = 1500

Calculating first the mean of squares:

rac{15889.5}{1500} ≈ 10.5923

Then, calculating the standard deviation:

$ext{Standard Deviation} ≈ \\sqrt{10.5923 - (3.2274)^2} ≈ \\sqrt{10.5923 - 10.4142} ≈ \\sqrt{0.1781} ≈ 0.4211$

Thus, the estimated standard deviation is approximately 0.42 kg.

To estimate the median weight, we first determine the cumulative frequency. The median is the 750th value (since 1500/2 = 750).

From the cumulative frequency:

• Less than 2.5kg: 67
• Less than 3.0kg: 347
• Less than 3.5kg: 1167

Since 750 falls between 3.0 kg and 3.5 kg, we interpolate:

Using the formula:

ext{Median} = L + rac{rac{N}{2} - CF}{f} imes c

Where:

• L = 3.0 (lower boundary of the median class)
• $N = 1500, CF = 347, f = 820 (frequency of median class), c = 0.5 (class width)$

Calculating:

ext{Median} = 3.0 + rac{750 - 347}{820} imes 0.5 ≈ 3.0 + rac{403}{820} imes 0.5 ≈ 3.0 + 0.245 ≈ 3.245

Thus, the estimated median birth weight is approximately 3.25 kg.

The distribution appears to be negatively skewed. This is determined by comparing the mean and median:

• The mean (approximately 3.23 kg) is less than the median (approximately 3.25 kg).

This typically indicates that the tail on the left side of the distribution is longer or fatter than the right side, leading to a negative skew.