The Lengths Of Pregnancies Are Normally Distributed

The Lengths Of Pregnancies Are Normally Distributed With Mean Μ 268

The lengths of pregnancies are normally distributed with mean µ = 268 days and standard deviation σ = 15 days. (a) If one pregnant woman is chosen at random, find the probability that her length of pregnancy is between 260 and 278 days. (b) Find the number of days above which lie the longest 1.5% of all pregnancies.

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The distribution of pregnancy lengths is a classic example of a continuous probability distribution modeled using the normal distribution. The use of the normal distribution in this context allows clinicians, researchers, and public health officials to assess probabilities related to pregnancy durations, which can be essential for planning and managing prenatal care. The parameters provided, with a mean of 268 days and a standard deviation of 15 days, serve as the basis for analyzing specific questions about pregnancy lengths.

Part (a) of the problem asks us to determine the probability that a randomly selected pregnancy lasts between 260 and 278 days. To address this, we first standardize the boundaries to the standard normal distribution. Standardization involves converting the raw scores (specific pregnancy lengths) into z-scores, which indicate how many standard deviations a value is from the mean. The formula used for this conversion is:

z = (X - µ) / σ

where X is the raw score, µ is the mean, and σ is the standard deviation.

Calculating the z-scores:

  • For X = 260 days: z = (260 - 268) / 15 = -8 / 15 ≈ -0.5333
  • For X = 278 days: z = (278 - 268) / 15 = 10 / 15 ≈ 0.6667

Next, these z-scores correspond to probabilities on the standard normal distribution. Using standard normal distribution tables or a calculator, we find the cumulative probabilities:

  • P(Z
  • P(Z

The probability that a pregnancy length falls between 260 and 278 days is then calculated by subtracting the lower cumulative probability from the upper:

P(260

Therefore, there is approximately a 45% chance that a randomly chosen pregnancy lasts between 260 and 278 days. This probability provides insight into the typical variation in pregnancy duration and can guide expectations for healthcare providers and expectant mothers.

Part (b) involves finding the cutoff point above which lie the longest 1.5% of pregnancies. This is a question of determining a high percentile in the normal distribution. Specifically, it requires identifying the z-score that corresponds to the top 1.5% of the distribution. Since the normal distribution is symmetric, this can be found by looking up the z-score associated with the 98.5th percentile (100% - 1.5%) on a standard normal table or calculator.

The z-score associated with the 98.5th percentile is approximately 2.17 (from standard normal distribution tables). Using the z-score formula to find the corresponding pregnancy length (X):

X = μ + zσ = 268 + (2.17)(15) ≈ 268 + 32.55 ≈ 300.55 days

This means that pregnancies longer than approximately 301 days (rounding to the nearest whole day) account for the longest 1.5% of all pregnancies. This information can be crucial for obstetricians and healthcare professionals to identify unusually prolonged pregnancies that may require special attention or intervention.

Understanding the probability and percentile calculations within a normal distribution context allows for better risk assessment and management of pregnancy-related health issues. Employing statistical tools such as z-scores and standard normal distribution tables enhances decision-making processes, ensuring that healthcare strategies are rooted in quantitative analysis.

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