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Consider babies born in the "normal" range of \(37-43\) weeks gestational age. The paper referenced in Example \(7.27\) ("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 \(\mu=3500\) grams and standard deviation \(\sigma=\) 600 grams is a reasonable model for the probability distribution of the continuous numerical variable \(x=\) 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 \(4000 \mathrm{~g}\) ? is between 3000 and \(4000 \mathrm{~g} ?\) b. What is the probability that the birth weight of a randomly selected full- term baby is cither less than \(2000 \mathrm{~g}\) or greater than \(5000 \mathrm{~g}\) ? 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? e. If \(x\) is a random variable with a normal distribution and \(a\) is a numerical constant \((a \neq 0)\), then \(y=a x\) also has a normal distribution. Use this formula to determine the distribution of full-term baby birth weight expressed in pounds (shape, mean, and standard deviation), and then recalculate the probability from Part (c). How does this compare to your previous answer?

Step by step solution

01

Find probability of birth weight exceeding 4000 grams

A Z-score should be calculated using the formula \(Z = (X - \mu) / \sigma = (4000 - 3500) / 600 \approx 0.833\) . Using a standard normal distribution (Z-table) to find the area to the right of this Z-score, gives the required probability.

02

Find probability of a birth weight being between 3000 and 4000 grams

First, calculate the Z-scores for 3000 and 4000 grams using \(Z = (X - \mu) / \sigma\) . Then use the standard normal distribution (Z-table) to correspond these Z-scores with the probabilities and find the difference between the two to obtain the probability of a birth weight being between 3000 and 4000 grams.

03

Compare probabilities of a birth weight less than 2000 grams or greater than 5000 grams

Similarly to steps 1 and 2, calculate the Z-scores for 2000 and 5000 grams and find the associated probabilities on the Z-table. Since these are two separate cases, add the two resulting probabilities.

04

Calculate probability of a birth weight exceeding 7 pounds

First, convert pounds into grams. 7 pounds is equivalent to approximately 3175.13 grams. Now calculate the Z-score for 3175.13, then find the proportion of the standard normal curve to the right of this Z-score – this is the required probability.

05

Determine distribution of birth weights in pounds

Apply \(\mu = E(ax) = aE(x)\) and \(\sigma = \sqrt{Var(ax)} = |a|\sqrt{Var(x)}\) to transform the birth weights from grams to pounds and recalculate the probability for a weight exceeding 7 pounds. Compare this probability with the one calculated in step 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

A probability distribution is like a map that represents all possible values a random variable can take and the likelihood of each value occurring. When we're talking about birth weights of babies born in the normal range of 37-43 weeks gestational age, we're looking at a continuous probability distribution. Specifically, it's a normal distribution with a mean and a standard deviation.
The mean is the average value, in this case, 3500 grams. The standard deviation, 600 grams, measures how spread out the numbers are around the mean. For a normal distribution like this, the famous "bell curve" depicts how birth weights cluster around the mean. Most babies will have weights near 3500 grams, but the curve tails off on either side, showing fewer occurrences as we move towards the less common weights.
To find the probability of a specific birth weight range, like exceeding 4000 grams, we'd use this distribution. This involves calculating a Z-score, which tells us how many standard deviations away from the mean our value is. Knowing this helps us find probabilities associated with different weights by looking them up on a standard normal distribution table.

Z-Score Calculation

Z-score calculation is a handy way to compare a specific weight to the overall distribution. In simpler terms, the Z-score tells us how unusual or typical a particular birth weight is in comparison to the average. The formula for calculating a Z-score is:

• \[ Z = \frac{X - \mu}{\sigma} \]

Here, \(X\) is the weight of the baby, \(\mu\) is the mean birth weight (3500 grams in our problem), and \(\sigma\) is the standard deviation (600 grams here).
Let's say we want to find out how common it is for a baby to weigh more than 4000 grams. We plug these values into the formula, and the result is our Z-score. A Z-score of 0 would mean the weight is exactly at the mean. Positive or negative Z-scores tell us how far from the mean a particular weight is, in terms of standard deviations. And by checking this score against a Z-table, we can figure out the probability or proportion of birth weights that are above, below, or between certain values.

Gestational Age

Gestational age refers to the length of time a baby spends developing in the womb, usually measured in weeks. Full-term gestation typically spans from 37 to 43 weeks. This time frame is crucial because the baby's development during this period can significantly impact their birth weight. Generally, a longer gestational age allows for more growth, potentially resulting in a higher birth weight.
However, even within this 'normal' gestational age range, there's variability. That's why statistics like the mean and standard deviation are important when talking about birth weights. They help us understand and predict the typical birth weight while acknowledging that each baby's development is unique. Understanding gestational age is essential for expecting parents, as it offers insight into what might be a healthy birth weight for their baby.

Birth Weight Analysis

Birth weight analysis involves examining the distribution and probabilities of various birth weights to understand patterns and predict outcomes. This type of analysis can inform healthcare decisions and parental expectations.
Looking at a normal distribution of birth weights, we examine extreme observations and ordinary occurrences. For instance, calculating the likelihood of a birth weight less than 2000 grams or greater than 5000 grams allows us to assess risks associated with significantly underweight or overweight newborns.
These probabilities hold clinical importance. Healthcare professionals use such analyses to determine if interventions might be needed to support the health of the mother and child. Furthermore, birth weight can serve as an indicator of a baby's immediate health needs or potential future health issues. Understanding both common and rare birth weights through rigorous analysis helps in crafting better healthcare strategies and ensuring the well-being of newborns.