The Distribution Of Total Body Protein In Adult Men

The Distribution Of Total Body Protein In Adult Men With Liver Cirrhos

The distribution of total body protein in adult men with liver cirrhosis is approximately normal with a mean of 9.8 kg and a standard deviation of 0.1 kg. The task is to determine the probability that a randomly selected man with liver cirrhosis has a total body protein between 9.75 kg and 9.85 kg.

Given that the distribution is normal, we can utilize properties of the normal distribution to compute the probability. First, we standardize the values to convert them into z-scores, which measure how many standard deviations each value is from the mean. The formula for the z-score is:

z = (X - μ) / σ

where X is the value in question, μ is the mean, and σ is the standard deviation.

Calculating the z-scores for the bounds:

• For the lower bound, X = 9.75 kg:
• z_lower = (9.75 - 9.8) / 0.1 = -0.05 / 0.1 = -0.5
• For the upper bound, X = 9.85 kg:
• z_upper = (9.85 - 9.8) / 0.1 = 0.05 / 0.1 = 0.5

Once the z-scores are known, we can consult the standard normal distribution table or use statistical software to find the probabilities associated with these z-values.

The probability that Z is between -0.5 and 0.5 can be expressed as:

P(-0.5 ≤ Z ≤ 0.5) = Φ(0.5) - Φ(-0.5)

where Φ(z) is the cumulative distribution function (CDF) for the standard normal distribution.

From standard normal distribution tables or calculators, we find:

• Φ(0.5) ≈ 0.6915
• Φ(-0.5) ≈ 1 - Φ(0.5) = 1 - 0.6915 = 0.3085

Therefore, the probability that the total body protein of a randomly selected man falls between 9.75 kg and 9.85 kg is:

0.6915 - 0.3085 = 0.3830

This indicates that there is approximately a 38.3% chance that a man with liver cirrhosis has total body protein in this range.

Note that the initial reference provided a probability of approximately 0.3173 for a similar range. The variation might be due to different calculation methods or rounding, but based on standard normal approximation, the result is approximately 0.3830.

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