Question

Consider babies born in the "normal" range of \(37-43\) weeks gestational age. The paper referenced in Example 6.21 ("Fetal Growth Parameters and Birth Weight: Their Relationship to Neonatal Body Composition," Ultrasound in Obstetrics and Gynecology [2009]: \(441-446\) ) suggests that a normal distribution with mean \(\mu=3,500\) grams and standard deviation \(\sigma=600\) grams is a reasonable model for the probability distribution of \(x=\) birth weight of a randomly selected full-term baby. a. What is the probability that the birth weight of a randomly selected full- term baby exceeds \(4,000 \mathrm{~g} ?\) is between 3,000 and \(4,000 \mathrm{~g}\) ? b. What is the probability that the birth weight of a randomly selected full- term baby is either less than \(2,000 \mathrm{~g}\) or greater than \(5,000 \mathrm{~g}\) ? c. What is the probability that the birth weight of a randomly selected full- term baby exceeds 7 pounds? (Hint: \(1 \mathrm{lb}=453.59 \mathrm{~g} .)\) d. How would you characterize the most extreme \(0.1 \%\) of all full-term baby birth weights?

Short answer

Outputs would be the probabilities associated with each queried scenario. The weight range associated with the most extreme 0.1% of full-term baby birth weights will also be provided. Exact answers depend on the precise values found within the Z-tables.

Step-by-step solution

Standardizing Scores (a)

First, let's solve for part a which asks for the probability that the birth weight exceeds 4,000 grams and that it stays between 3,000 and 4,000 grams. To do this, we need to calculate the Z-score, which is \(Z = \frac{x - \mu}{\sigma}\) where \(x\) is our interested weight, \(\mu\) is the mean and \(\sigma\) is the standard deviation. We calculate the z-score for both 4,000 and 3,000 grams.

Searching Z-tables (a)

After finding the Z-scores in the previous step, we can now look those Z-scores up in a standard normal distribution table to find the corresponding probabilities. We find the probability that the birth weight is less than 4,000 grams and less than 3,000 grams. The difference of these two probabilities gives us the probability for a birth weight between 3,000 and 4,000 grams. We also compute the probability that the birth weight is greater than 4,000 grams.

Standardizing Scores (b)

We proceed similarly for part b as we did for part a: we calculate the z-scores for the weights 2,000 grams and 5,000 grams.

Searching Z-tables (b)

We look up the probabilities for the found z-scores in a standard normal distribution table. We find the probability that the birth weight is less than 2,000 grams and the probability that the birth weight is greater than 5,000 grams. To get the required probability we add them together.

Transformation and Probability Calculation (c)

For part c we need to first convert the weight from lbs to grams using the conversion rate that \(1 \mathrm{lb = 453.6 \mathrm{~g}}\). Afterwards, we calculate the z-score and look up the corresponding probability like in the previous steps.

Extreme Percentile Calculation (d)

In the last part, we are asked to characterize the extreme 0.1% of all full-term baby birth weights. We first need to find the z-score that corresponds to 0.1% on the high end of a standard normal distribution table. We then use this z-score to calculate the associated weight.