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Birth weight is a crucial indicator of a newborn's health. Medical professionals often use birth weight statistics to assess and predict health outcomes. Typically, birth weight data is normally distributed, meaning it forms a bell-shaped curve when plotted. This makes it easier to use statistical methods to determine probabilities and significance levels.

In the provided example, the mean birth weight is given as 3152.0 grams, with a standard deviation of 693.4 grams. These values are essential for understanding the distribution and making further calculations. For instance:

• The mean (\( \mu \)) represents the average birth weight in the dataset.
• The standard deviation (\( \sigma \)) shows the variation or dispersion of the birth weights around the mean.

By understanding these basic statistics, we can assess scenarios like determining the likelihood of a baby having a 'low birth weight,' which is defined as less than 2495 grams in this case. Using these statistical measures, we can proceed to more detailed calculations such as the Z-score and probability.

A Z-score is a measurement that describes a value's position relative to the mean of a group of values. It is measured in terms of standard deviations from the mean. To calculate the Z-score for a given birth weight, use the formula: \[ Z = \frac{(X - \mu)}{\sigma} \] where:

• \( X \) is the value we are interested in (in this case 2495 grams).
• \( \mu \) is the mean (3152.0 grams).
• \( \sigma \) is the standard deviation (693.4 grams).

Plugging in the numbers: \[ Z = \frac{(2495 - 3152.0)}{693.4} \approx -0.9486 \] This Z-score tells us how many standard deviations a birth weight of 2495 grams is from the mean. A Z-score of approximately -0.9486 indicates that 2495 grams is 0.9486 standard deviations below the mean birth weight. Knowing the Z-score, we can now use the Z-table to find the probability associated with this Z-value.

Probability measures the likelihood of a certain event occurring. In the context of birth weights, we can calculate the probability that a randomly selected birth weight is below a certain value, such as 2495 grams.

For a Z-score of -0.9486, we use the Z-table to find the corresponding probability. The Z-table gives us the area to the left of a given Z-score on the standard normal distribution. For \( Z = -0.9486 \), the probability is approximately 0.1713. Thus, there is a 17.13% chance that a randomly selected birth weight will be less than 2495 grams.

On the other hand, when determining significantly low birth weights defined by a particular probability, say 0.05, we refer to the Z-table to find the corresponding Z-score, which is -1.645. We then convert this Z-score back to the original weight using the formula: \[ X = \mu + Z \times \sigma \] Calculating this gives us: \[ X = 3152.0 + (-1.645) \times 693.4 \approx 2011.97 \] grams. Consequently, any birth weight less than 2011.97 grams is considered significantly low with a probability of 0.05 or less. Understanding these probabilities helps in making informed medical decisions.