91Ó°ÊÓ

The standard normal distribution is a special normal distribution. It has a mean of 0 and a standard deviation of 1. It's used to simplify finding probabilities for any normal distribution. When we talk about a normal distribution, we often deal with different means and standard deviations. The standard normal distribution standardizes this by converting any normal random variable to one that follows this special distribution.

• Mean (bc) of 0
• Standard deviation (c3) of 1

This conversion to a standard normal variable uses the z-score formula:\[ z = \frac{x - \mu}{\sigma}\]Here, any normal random variable can be described in terms of the standard normal distribution. By doing so, it allows us to use standard tables or calculators for finding probabilities. This transformation is critical when working with any normal data and forms the basis of many statistical methods.

Cumulative probability refers to the probability that a random variable will have a value less than or equal to a specified value. In the context of the standard normal distribution, it is the area under the curve to the left of a given z-score.When we find cumulative probability:

• We increase from left to right.
• We consider all probabilities adding up from the start point.

In exercises requiring cumulative probability, a lookup table or statistical software can determine this value quickly. For example, if we are looking for probabilities in a right tail, as we did in our solution to find \( P(x \geq 90) \), we actually converted it to finding a cumulative probability for a z-score of \(-0.67\) and then adjusted by subtracting from one. This is due to the properties of the distribution curve being symmetrical.

The z-score is a measure that describes a value's relation to the mean of a group of values. In other words, it's a way to understand how far or how many standard deviations a data point is from the mean. The formula for the z-score is:\[z = \frac{x - \mu}{\sigma} \]This simple formula standardizes any variable to fit the standard normal distribution. By transforming a normal value to a z-score, we use a universal scale.

• A z-score of 0 means the data point is exactly at the mean.
• Positive z-scores represent values above the mean.
• Negative z-scores represent values below the mean.

In our calculation, we found \( z = -0.67 \), indicating that 90 is two-thirds of a standard deviation below the mean. This standardization is key for comparing values from different distributions or for calculating probabilities using the standard normal distribution.