Final answer:
The 85th percentile for baby weights is approximately 16.32 lbs, the 14th percentile is about 7.77 lbs, and the first quartile is around 9.43 lbs, calculated using z-scores and the standard normal distribution.
Step-by-step explanation:
To find the percentile of the baby weights, we use the provided average weight and standard deviation with a Cumulative Normal Distribution Table. The process involves converting the percentile to a z-score and then using the formula:
Z = (X - μ) / σ
Where Z is the z-score, X is the weight, μ is the mean, and σ is the standard deviation.
(a) 85th percentile
First, we locate the z-score that corresponds to the 85th percentile in the Cumulative Normal Distribution Table, which is approximately 1.04. Using the formula with our mean (μ = 12.1 lbs) and standard deviation (σ = 4.1 lbs), we calculate:
X = Zσ + μ = 1.04(4.1 lbs) + 12.1 lbs ≈ 16.324 lbs, or about 16.32 lbs when rounded to two decimal places.
(b) 14th percentile
Similarly, we find the z-score for the 14th percentile, which is approximately -1.08. Thus:
X = Zσ + μ = -1.08(4.1 lbs) + 12.1 lbs ≈ 7.772 lbs, or approximately 7.77 lbs after rounding.
(c) First quartile
The first quartile, or 25th percentile, corresponds to a z-score of about -0.675. Therefore:
X = Zσ + μ = -0.675(4.1 lbs) + 12.1 lbs ≈ 9.425 lbs, which rounds to approximately 9.43 lbs.