Find The Probability That A Random Man Is S

Find the probability that if an individual man is randomly selected, his weight will be greater than 180 lb.

The problem states that men’s weights are normally distributed with a mean (μ) of 172 pounds and a standard deviation (σ) of 29 pounds. To find the probability that a randomly selected man weighs more than 180 pounds, we first calculate the z-score for 180 pounds using the formula:

z = (X - μ) / σ

Substituting the given values:

z = (180 - 172) / 29 ≈ 8 / 29 ≈ 0.2759

Using standard normal distribution tables or a calculator, we find the probability P(Z > 0.2759). Since standard normal tables typically give P(Z

P(Z > 0.2759) = 1 - P(Z

Looking up P(Z

1 - 0.608 = 0.392

Thus, there is approximately a 39.2% chance that a randomly selected man weighs more than 180 pounds.

Paper For Above instruction

The distribution of men's weights follows a normal distribution characterized by a mean of 172 pounds and a standard deviation of 29 pounds. To determine the likelihood of selecting a man weighing more than 180 pounds, we employ statistical inference methods rooted in the properties of normal distributions. First, the z-score calculation transforms the specific weight into a standardized value indicating how many standard deviations it is from the mean:

z = (X - μ) / σ = (180 - 172) / 29 ≈ 0.2759.

Next, referencing the standard normal distribution table or employing computational tools, we find the cumulative probability up to this z-score. The area under the normal curve to the left of z = 0.2759 corresponds approximately to 0.608. Since we are interested in the probability exceeding 180 pounds, we compute the complement:

P(X > 180) = 1 - P(Z

This indicates that there is a roughly 39.2% chance that a randomly selected man weighs more than 180 pounds. This probability is essential for health assessments, marketing strategies, or nutritional planning where weight distributions are relevant.

Normal distribution analysis is vital in understanding and managing populations where variability plays a critical role. The probability derived here offers insights into weight-related health risks and resource allocation, demonstrating the utility of statistical tools in practical decision-making.

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