Heights of women are normally distributed with mean 63.7 inches and standard deviation 2.47 inch

step 1 In this question, we are told that the heights of women are normally distributed with a mean of

Heights of women are normally distributed with mean 63.7...

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean. It is characterized by its bell-shaped curve, where most of the data points cluster around the mean, and the probabilities for values further away from the mean taper off equally in both directions. Its two parameters, the mean and the standard deviation, determine its location and spread, respectively, making it a common model for naturally occurring phenomena.

Percentile

A percentile is a statistical measure that indicates the value below which a given percentage of observations in a data set falls. For example, the 10th percentile is the value below which 10% of the observations lie, helping to identify the relative standing of a data point within a distribution. Percentiles are used to understand the distribution of data, compare individual scores within a dataset, and determine thresholds for classification.

Z-Score and Standardization

The Z-score is a dimensionless metric that indicates how many standard deviations an element is from the mean of its distribution. Standardization is the process of converting a normal variable into a standard normal variable (commonly referred to as the Z-distribution) by subtracting the mean and dividing by the standard deviation. This procedure facilitates the use of standard normal distribution tables or software functions to calculate probabilities and percentiles in various contexts.

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