91Ó°ÊÓ

The z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. When it comes to the normal distribution, it helps us understand where a particular point falls on the standard bell curve. The formula for calculating the z-score is given by:

• Formula: \( z = \frac{x - \mu}{\sigma} \)
• Where: \(x\) is the value in question, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

To find how far away a number is from the average, compared to the overall spread, compute the z-score. It tells us how many standard deviations an element is from the mean.
For example, in the problem at hand, the z-score came out to be approximately \(-0.67\), indicating that 90 is less than one standard deviation below the mean of 100.

The standard normal distribution, often called the "bell curve," is a special case of the normal distribution. It has a mean (bc) of 0 and a standard deviation (c3) of 1. This distribution is crucial for calculating probabilities and is symmetric about the mean.
When you convert a normal distribution to a standard normal distribution using the z-score formula, you are essentially mapping the data onto this special curve. This allows you to use standardized tables to find probabilities easily. Each z-score represents a point along the x-axis of this curve.
The standard normal distribution table, sometimes called the "z-table," helps in finding the percentage of values below (or sometimes above) a given z-score. Such tables display the cumulative probability associated with that score. In our example, the z-score \(-0.67\) showed us that about 25.14% of the data falls below that point.

Cumulative probability is a way of expressing the probability that a random variable is less than or equal to a certain value. Using the standard normal distribution table, you gain insights into this probability over a range of values.
By looking up a z-score in a z-table, we're actually finding the cumulative probability – the total probability up to that point. This is pivotal when calculating probabilities for events linked with a normal distribution.

• In this problem, we found that the cumulative probability for \(z = -0.67\) is approximately 0.2514.
• This means that 25.14% of the distribution lies below the score in question.

However, since we wanted to know the probability for \(x \geq 90\), we worked with one minus this cumulative probability to get 0.7486, demonstrating the complement rule in probability. This means 74.86% of the data falls above the score of 90.