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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
Question 5
A midwife records the weights, in kg, of a sample of 50 babies born at a hospital. Her results are given in the table below. Weight (w kg) Frequency (f) Weig... show full transcript
Worked Solution & Example Answer: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
Step 1
Calculate the width and height of the bar representing a weight of 3 ≤ w < 3.5.
Answer
The width of the bar is the difference between the upper and lower limits of the interval.
Width:
The height can be found using the frequency of the interval.
Using the frequency density: ext{Height} = rac{ ext{Frequency}}{ ext{Width}} = rac{17}{0.5} = 34 ext{ cm}
Step 2
Use linear interpolation to estimate the median weight of these babies.
Answer
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
Step 3
Step 4
Find an estimate of the standard deviation of the weights of these babies.
Answer
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:
After calculations, we estimate:
ext{Standard Deviation} acksimeq 0.680
Step 5
Step 6
With reference to your answers to (b), (c)(i) and (c)(ii), comment on Shyam's decision.
Answer
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.
Step 7
Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the mean.
Answer
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.
Step 8
Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the standard deviation.
Answer
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.