How tall? 6.2 ) The heights of young men follow a Normal...

How tall? 6.2 ) The heights of young men follow a Normal distribution with mean 69.3 inches and standard deviation 2.8 inches. The heights of young women follow a Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. (a) Let $M=$ the height of a randomly selected young man and $W=$ the height of a randomly selected young woman. Describe the shape, center, and spread of the distribution of $M-W$ (b) Find the probability that a randomly selected young man is at least 2 inches taller than a randomly selected young woman. Show your work.

How tall? 6.2 ) The heights of young men follow a Normal distribution with mean 69.3 inches and standard deviation 2.8 inches. The heights of young women follow a Normal distribution with mean 64.5 inches and standard deviation 2.5 inches.
(a) Let $M=$ the height of a randomly selected young man and $W=$ the height of a randomly selected young woman. Describe the shape, center, and spread of the distribution of $M-W$
(b) Find the probability that a randomly selected young man is at least 2 inches taller than a randomly selected young woman. Show your work.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve where most of the observations cluster around the central peak, and the probabilities for values further from the mean decrease exponentially. It is defined by two parameters: the mean, which indicates the center of the distribution, and the standard deviation, which measures the spread or dispersion of the data around the mean.

Difference of Independent Normal Random Variables

When two independent random variables, each following a normal distribution, are subtracted, the resulting variable also follows a normal distribution. The mean of the difference is the difference of the means of the original variables, and the variance is the sum of their variances. This property allows for the analysis of differences between groups or measurements.

Standardization

Standardization is the process of converting a normal random variable into a standard normal variable (z-score) by subtracting the mean and dividing by the standard deviation. This transformation facilitates the use of standard normal tables or computational tools to calculate probabilities, as it normalizes the distribution to have a mean of 0 and a standard deviation of 1.

Probability Calculation Using Z-Scores

Once a random variable is standardized to a z-score, probability calculations can be easily performed by referring to the cumulative distribution function of the standard normal distribution. This method provides a systematic approach for determining the likelihood of observing a value above or below a specified threshold.

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