Scores on the ACT college entrance exam follow a bell-shaped...

Scores on the ACT college entrance exam follow a bell-shaped distribution with mean 18 and standard
deviation 6. Wayne’s standardized score on the ACT was ?0.7. What was Wayne’s actual ACT score?
(a) 4.2 (c) 13.8 (e) 22.2
(b) ?4.2 (d) 17.3

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Key Concepts

Normal Distribution

A normal distribution is a probability distribution that is symmetric and bell-shaped, where most values cluster around a central mean. This distribution is defined by its mean and standard deviation, which describe the center and spread of the data, making it widely applicable in statistics and various fields for representing real-world phenomena.

Standard Score (Z-Score)

A z-score, or standardized score, measures how many standard deviations an individual data point is from the mean of its distribution. It allows comparisons between different data sets or distributions by normalizing values, thus indicating the relative position of a score within the overall distribution.

Standardization Formula

The standardization formula X = ? + z? is used to convert a z-score back into its original raw score. Here, X represents the actual score, ? is the mean, ? is the standard deviation, and z is the standardized score. This formula is essential for translating statistical results back into the original measurement scale.

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