X is a normally distributed random variable with mean -25...

$X$ is a normally distributed random variable with mean -25 and standard deviation $4 .$ Find the probability indicated.
a. $\quad P(X<-27.2)$
b. $\quad P(X<-14.8)$
c. $\quad P(X>-33.1)$
d. $\quad P(X>-16.5)$

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (which indicates the center of the distribution) and its standard deviation (which measures the dispersion or spread of the distribution). It is widely used in statistics and many real-world applications because numerous natural phenomena tend to follow this pattern.

Standardization

Standardization is the process of converting a normally distributed variable into a standard normal variable (usually denoted as Z), which has a mean of 0 and a standard deviation of 1. This is done using the transformation Z = (X - ?) / ?, where X is the original variable, ? is the mean, and ? is the standard deviation. This allows for the use of standard normal distribution tables or computational tools to find probabilities.

Z-Score

A Z-score represents the number of standard deviations a data point is from the mean. It is a crucial concept in probability and statistics because it enables the comparison of scores from different normal distributions. By converting any normal variable to a Z-score, we obtain a common scale for evaluating probabilities and percentiles.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable gives the probability that the variable will take a value less than or equal to a specific value. For the standard normal distribution, the CDF is used to determine probabilities for any value after standardization. It is central in calculating probabilities, such as finding the area under the curve to the left of a given Z-score.

Tail Probabilities

Tail probabilities refer to the probabilities found in the extreme ends (left-tail or right-tail) of the distribution. For a normally distributed variable, these represent the chances that the observed value is either significantly lower or significantly higher than the mean. Calculation of these requires either direct use of the CDF or the complementary probability (1 - CDF), depending on whether the tail of interest is the left or right one.

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