In each case, determine the value of the constant c that...

In each case, determine the value of the constant $c$ that makes the probability statement correct.
(a) $\Phi(c)=.9838$
(b) $P(0 \leq Z \leq c)=.291$
(c) $P(c \leq Z)=.121$
(d) $P(-c \leq Z \leq c)=.668$
(e) $P(c \leq|Z|)=.016$

Key Concepts

Absolute Value and One-sided/Tail Probabilities

When probability statements involve absolute values, such as P(|Z|?c) or P(c ? |Z|), the calculations involve both tails of the normal distribution. The symmetry of the distribution is used to appropriately allocate the given probability to each tail, thereby aiding in determining the necessary z-value thresholds.

Probability Intervals under the Normal Curve

Calculating probabilities over intervals in the standard normal distribution involves integrating the probability density function or using the difference between cumulative probabilities. This concept is crucial when determining values like the constant c in statements about the probability that the variable falls between or outside certain bounds.

Symmetry of the Normal Distribution

The symmetry of the standard normal distribution about zero implies that the probabilities associated with z values of equal magnitude but opposite signs are identical. This property is used to simplify probability calculations, especially when dealing with intervals or absolute values, as it allows one to equate probabilities on one side of the mean with those on the other side.

Inverse CDF (Quantile Function)

The inverse CDF, also known as the quantile function, is used to determine the z value corresponding to a given cumulative probability. This is particularly important when solving for constants in probability statements, such as determining z values from specified tail probabilities or central probabilities.

Cumulative Distribution Function (CDF)

The CDF of a random variable gives the probability that the variable takes on a value less than or equal to a specified value. In the context of the standard normal distribution, the function ?(z) is used to denote the probability that Z is less than or equal to a given z value.

Standard Normal Distribution

The standard normal distribution is a fundamental probability distribution in statistics characterized by a mean of 0 and a standard deviation of 1. It serves as the basis for many inferential techniques and is often used to model natural phenomena. Z-scores represent a data point's distance from the mean in units of standard deviations.

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