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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 Given Problem

We are given that the birth weight is normally distributed with a mean of \(3432\, \text{g}\) and a standard deviation of \(482\, \text{g}\). We need to calculate probabilities for various conditions and part (e) involves converting the distribution from grams to pounds.

02

Convert Pounds to Grams (Part c)

Since \(1 \text{lb} = 453.592\, \text{g}\), 7 \(\text{lb}\) is equivalent to \(3174.144\, \text{g}\). We will calculate the probability that a baby's weight exceeds this value.

03

Utilize Standard Normal Distribution (Part a)

1. To find the probability of exceeding \(4000\, \text{g}\), use the formula: \[ Z = \frac{X - \mu}{\sigma} \] where \(X = 4000\, \text{g}\), \(\mu = 3432\, \text{g}\), and \(\sigma = 482\, \text{g}\). Calculate \(Z\) and find the probability using the standard normal table.2. For the probability between \(3000\, \text{g}\) and \(4000\, \text{g}\), calculate \(Z\) values for both \(3000\, \text{g}\) and \(4000\, \text{g}\), then find the difference in probabilities using the normal distribution table.

04

Probability of Extreme Values (Part b)

For less than \(2000\, \text{g}\) and greater than \(5000\, \text{g}\), calculate \(Z\) values:1. \(Z\) for \(2000\, \text{g}\).2. \(Z\) for \(5000\, \text{g}\).Sum the respective probabilities to find the total probability.

05

Determine the Most Extreme 0.1% (Part d)

To find the extreme 0.1%, use the \(Z\)-score corresponding to 0.999 (because this folds the lowest 0.1% on the left tail). This is found using the normal distribution table, allowing retrieval of the equivalent weight value using the formula: \[ X = \mu + \sigma \cdot Z \] where the calculated \(Z\) is used.

06

Convert Distribution to Pounds (Part e)

Use the conversion \(a = \frac{1}{453.592}\) since there are 453.592 grams in a pound. With \(Y = a \cdot X\) and \(X\sim N(3432, 482^2)\), \(Y\) also has a normal distribution:- Mean: \(\mu_Y = a\mu_X = 3432 \times \frac{1}{453.592}\).- Std Dev: \(\sigma_Y = a\sigma_X = 482 \times \frac{1}{453.592}\).

07

Recalculate Probability in Pounds (Part c)

With distribution expressed in pounds, recalculate the probability that a baby's weight exceeds 7 pounds using the conversion from the previous step and compare with previous results from part c.

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Key Concepts

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

Normal Distribution

The concept of normal distribution is fundamental in statistics, describing how the values of a dataset are spread. It is often referred to as a "bell curve" because of its symmetrical shape when plotted. The mean, median, and mode of a normal distribution are all the same. This statistical tool is widely used to represent real-world data that clusters around a central value.

• **Symmetrical Shape:** The normal distribution graph is symmetrical about the mean, indicating that data points are evenly distributed on both sides.
• **Mean as the Center:** In a normal distribution, most of the data points are situated close to the mean, with fewer points occurring as you move further away.
• **Distribution Characteristics:** Understanding a normal distribution helps predict the likelihood of different outcomes and estimate where most data is likely to fall.

For the birth weight of newborns, the data is assumed to be normally distributed, with a specific mean and standard deviation, providing a framework for probability calculations.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the individual data points in a dataset deviate from the mean. A higher standard deviation indicates that the data points are spread out over a wider range of values, whereas a lower standard deviation shows that they are closer to the mean.

• **Calculation:** The standard deviation is calculated as the square root of the variance, with variance being the average of the squared differences from the mean.
• **Role in Normal Distribution:** In a normal distribution, approximately 68% of data falls within one standard deviation of the mean, while about 95% falls within two standard deviations.
• **Significance:** Understanding standard deviation is essential for interpreting data variability and applying the normal distribution model accurately.

In our exercise, the standard deviation of newborn birth weights is 482 grams. This helps us understand the spread of birth weights around the mean of 3432 grams.

Z-Score

The Z-score is a statistical measure that describes a data point's relationship to the mean of a group of points. It is expressed in terms of standard deviations from the mean. Using Z-scores makes it easier to understand how far or close a value is compared to the average of a dataset.

• **Formula:** The Z-score formula is: \[ Z = \frac{X - \mu}{\sigma} \] where \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
• **Interpretation:** A Z-score of 0 indicates that the data point is exactly at the mean, while a positive Z-score means it is above the mean, and a negative Z-score signifies it is below the mean.
• **Applications:** Z-scores allow us to calculate the probability of a score occurring within a normal distribution and compare scores from different datasets.

In the problem stated, Z-scores help us find the probabilities of different birth weights, such as the likelihood of a baby weighing more than 4000 grams.

Conversion Factors

Conversion factors are used to change units from one measurement system to another, maintaining the same quantity despite the unit change. In the context of our problem, we need conversion factors to convert weights from grams to pounds, as birth weights are sometimes recorded using different units.

• **Unit Conversion:** For weight, the conversion factor from grams to pounds is that 1 pound equals approximately 453.592 grams.
• **Application:** Conversion factors are crucial when recalculating probabilities after converting a dataset to a different unit of measure.
• **Impact on Normal Distribution:** When converting units, the mean and standard deviation of the distribution must also be converted using the same factor to keep the data consistent.

In our exercise, this conversion is essential for recalculating the distribution and probability questions in the pound unit, ensuring accurate and comparable results across different systems.