Heights of Men in the US Heights of adult males in the US...

Heights of Men in the US Heights of adult males in the US are approximately normally distributed with mean 70 inches $(5$ ft 10 in $)$ and standard deviation 3 inches. (a) What proportion of US men are between $5 \mathrm{ft}$ 8 in and $6 \mathrm{ft}$ tall $(68$ and 72 inches, respectively)? (b) If a man is at the 10 th percentile in height, how tall is he?

Heights of Men in the US Heights of adult males in the US are approximately normally distributed with mean 70 inches $(5$ ft 10 in $)$ and standard deviation 3 inches.
(a) What proportion of US men are between $5 \mathrm{ft}$ 8 in and $6 \mathrm{ft}$ tall $(68$ and 72 inches, respectively)?
(b) If a man is at the 10 th percentile in height, how tall is he?

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) calculates the probability that a normally distributed random variable will have a value less than or equal to a specific value. In the context of the normal distribution, it is used to determine the proportion of observations below a given threshold once the corresponding z-score is known.

Percentiles

Percentiles are used to indicate the relative standing of a data point within a distribution. They represent the percentage of the data that falls below a given value. Calculating percentiles in the context of a normal distribution typically involves finding the z-score corresponding to the desired cumulative probability and then transforming back to the original scale.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric around its mean, with a bell-shaped curve. It is fully defined by its mean and standard deviation and is often used to model natural phenomena, such as heights, test scores, and measurement errors.

Standardization (Z-score)

Standardization involves converting a raw score from a normal distribution into a z-score by subtracting the mean and dividing by the standard deviation. This transformation produces a standard normal distribution with a mean of 0 and a standard deviation of 1, which allows for the use of standardized tables or software to find probabilities.

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