Find the value of x^* that yields the probability shown,...

Find the value of $x^{*}$ that yields the probability shown, where $X$ is a normally
distributed random variable $X$ with mean 83 and standard deviation $4 .$
a. $D\left(X<x^{4}\right)=0.87 \mathrm{CD}$
b. $P\left(X>x^{4}\right)=0.0800$

Key Concepts

Percentiles and Cumulative Distribution Function (CDF)

Percentiles indicate the value below which a given percentage of observations in a group falls. The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes a value less than or equal to a specified number. By using the CDF, one can determine the percentile corresponding to that value and vice versa.

Inverse Cumulative Distribution Function (Quantile Function)

The inverse cumulative distribution function, or quantile function, is used to determine the value of the variable corresponding to a specified probability in the distribution. It effectively 'inverts' the CDF by providing the required threshold such that the probability of observing a value below (or above) it equals a given value.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is determined by two parameters: the mean (?) and the standard deviation (?), which indicate the center and the spread of the distribution, respectively.

Z-Score Standardization

Z-score standardization is a process of converting a normally distributed variable to a standard normal variable (with mean 0 and standard deviation 1) using the formula z = (x - ?) / ?. This conversion simplifies the process of finding probabilities and percentiles, as standard normal distribution tables or functions can be used.

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