91Ó°ÊÓ

Consider babies born in the "normal" range of \(37-43\) weeks gestational age. Extensive data supports the assumption that for such babies born in the United States, birth weight is normally distributed with mean \(3432 \mathrm{~g}\) and standard deviation \(482 \mathrm{~g}\). [The article "Are Babies Normal?" (The American Statistician, 1999: 298–302) analyzed data from a particular year; for a sensible choice of class intervals, a histogram did not look at all normal, but after further investigations it was determined that this was due to some hospitals measuring weight in grams and others measuring to the nearest ounce and then converting to grams. A modified choice of class intervals that allowed for this gave a histogram that was well described by a normal distribution.] a. What is the probability that the birth weight of a randomly selected baby of this type exceeds \(4000 \mathrm{~g}\) ? Is between 3000 and \(4000 \mathrm{~g}\) ? b. What is the probability that the birth weight of a randomly selected baby of this type is either less than \(2000 \mathrm{~g}\) or greater than \(5000 \mathrm{~g}\) ? c. What is the probability that the birth weight of a randomly selected baby of this type exceeds \(7 \mathrm{lb}\) ? d. How would you characterize the most extreme . \(1 \%\) of all 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 to determine the distribution of 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

Understanding the Problem

We are given that birth weights are normally distributed with mean \(\mu = 3432\) g and standard deviation \(\sigma = 482\) g. We need to solve the following problems using this distribution.

02

Calculating Probability Exceeds 4000g (Part a)

First, we find the z-score for 4000 g using \( z = \frac{x - \mu}{\sigma} \): \[ z = \frac{4000 - 3432}{482} \approx 1.1785 \] Using the standard normal distribution table, the probability \(P(Z > 1.1785)\) is about 0.119. Therefore, the probability that the birth weight exceeds 4000 g is 0.119.

03

Calculating Probability Between 3000g and 4000g (Part a)

Calculate z-scores: For 3000 g, \( z = \frac{3000 - 3432}{482} \approx -0.895 \) For 4000 g, \( z = 1.1785 \) (as calculated before).Using the z-table, \( P(Z < 1.1785) \approx 0.8809 \) and \( P(Z < -0.895) \approx 0.1851 \). Therefore, the probability between these weights is \( 0.8809 - 0.1851 = 0.6958 \).

04

Probability Less Than 2000g or Greater Than 5000g (Part b)

Calculate z-scores:For 2000 g, \( z = \frac{2000 - 3432}{482} \approx -2.97 \). For 5000 g, \( z = \frac{5000 - 3432}{482} \approx 3.26 \).Using the z-table, \(P(Z < -2.97) \approx 0.0015\) and \(P(Z > 3.26) \approx 0.0006\).The combined probability is \(0.0015 + 0.0006 = 0.0021\).

05

Probability Exceeds 7 Pounds (Part c)

First, convert 7 pounds to grams: 7 lb \(= 3175.147\) g.Find z-score for 3175.147 g: \[ z = \frac{3175.147 - 3432}{482} \approx -0.532 \]Using the z-table, \(P(Z > -0.532) \approx 0.702 \). Thus, the probability that birth weight exceeds 7 lb is 0.702.

06

Characterizing Most Extreme 1% (Part d)

Find the z-score for the 99th percentile (0.99): \( z \approx 2.33 \) (from z-table).Calculate the birth weight threshold using: \[ x = \mu + z \sigma = 3432 + 2.33 \times 482 \approx 4555 \] g.Thus, the most extreme 1% of birth weights are greater than 4555 g.

07

Normal Distribution of Weights in Pounds (Part e)

Let \(Y = \frac{X}{453.592} \), converting g to lbs. Then, \( Y \sim N\left( \frac{3432}{453.592}, \frac{482}{453.592} \right) \approx N(7.57, 1.06) \).Recalculate probability from part c: Convert 7 lbs directly and calculate, which reconfirms \(P(Z > -0.534) \approx 0.702 \).

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Key Concepts

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

Birth Weight

Birth weight is a crucial indicator of a newborn's health and development. For babies born within the normal gestational range of 37 to 43 weeks, birth weight follows a normal distribution, which is a common continuous probability distribution. In the context of birth weight, a normal distribution suggests that most babies will have weights around the average, with fewer babies weighing much less or much more than the average.

• The average or mean birth weight is often around 3432 grams for babies in the United States.
• The variability or spread of this weight is represented by the standard deviation, which is 482 grams in this case.

Normal distribution is an important tool in statistics because it can describe many natural phenomena, and birth weight is no exception. By understanding this distribution, parents and healthcare professionals can monitor and assess the growth and health of newborns more effectively.

Probability

Probability is the measure of the likelihood of a certain event occurring. In statistics, especially when dealing with a normal distribution, we often want to calculate the probability of a particular outcome based on the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the distribution. For birth weights, this involves calculating the likelihood that a baby's weight falls into a specific range.Here's how we use probability with birth weights:

• To find the probability that a baby weighs more than a certain value (e.g., 4000 grams), we calculate the z-score and use standard normal distribution tables or software to find the probability associated with that z-score.
• To determine the probability between two weight values, like 3000 grams to 4000 grams, we find the probabilities of both ends using z-scores and subtract to find the area between them.
• The normal distribution also helps in estimating extreme probabilities, such as weights less than 2000 grams or more than 5000 grams, which are less common and often concern healthcare professionals.

Using these probabilities allows for making informed decisions regarding newborn care and understanding population health trends.

Z-Scores

Z-scores are a way of standardizing scores on a distribution. They indicate how many standard deviations an element is from the mean. When evaluating a normally distributed variable like birth weight, z-scores are essential for comparing different datas.Calculation and Interpretation:

• A z-score is calculated using the formula: \[ z = \frac{X - \mu}{\sigma} \]where \(X\) is the value in question, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
• A positive z-score indicates the value is above the mean, while a negative z-score reveals it's below the mean.
• Z-scores allow us to find probabilities using standard normal distribution tables, which contain cumulative probabilities for standard normal distributions.

By using z-scores, we translate specific birth weights into probabilities, facilitating an understanding of how common or rare a particular weight is in the population. This application is crucial for health assessments and statistical analysis.