A continuous random variable X has a normal distribution...

A continuous random variable $X$ has a normal distribution with mean 73 . The probability that $X$ takes a value greater than 80 is $0.212 .$ Use this information and the symmetry of the density function to find the probability that $X$ takes a value less than $66 .$ Sketch the density curve with relevant regions shaded to illustrate the computation.

Key Concepts

Area Under the Curve and Graphical Representation

In the context of a normal distribution, probabilities are calculated as areas under the curve of the density function. The graph of the normal curve is used to visually represent these probabilities, with areas corresponding to probabilities for different ranges of the variable. Shading these relevant regions in a sketch helps illustrate how the areas (and thus probabilities) are determined, reinforcing the connection between the algebraic computation of probabilities and their geometric interpretation.

Symmetry of the Normal Distribution

The symmetry of the normal distribution means that its left and right halves mirror each other about the mean. This property implies that the probability of a value deviating a certain distance below the mean is the same as the probability of deviating the same distance above the mean. This symmetry is particularly useful when calculating tail probabilities or comparing probabilities on either side of the mean.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped density curve which is symmetric about its mean. It is widely used in statistics due to the central limit theorem, which states that sums of random variables tend to have a normal distribution regardless of the original distribution. This concept helps in understanding how probabilities are distributed across different values of the variable.

Mean and Standard Deviation

The mean is the central value of the distribution and determines the location of the center of the density curve, while the standard deviation quantifies the spread or dispersion of the distribution about the mean. Together, these parameters fully characterize a normal distribution, allowing for the calculation of probabilities associated with different intervals on the distribution.

Standardization (Z-scores)

Standardization is the process of converting a value from a normal distribution to a standard normal distribution by subtracting the mean and dividing by the standard deviation. This transformation results in a z-score, a dimensionless number that indicates how many standard deviations the original value is from the mean. This is particularly useful for comparing different values and using standard normal tables to find probabilities.

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