91Ó°ÊÓ

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. The z-score is measured in terms of standard deviations from the mean. For example, a z-score of 1.0 means that the data point is one standard deviation away from the mean. Z-scores are a way of standardizing data points for purposes of comparison. It provides a way to compare different data points across different normal distributions.

• A positive z-score: Indicates the data point is above the mean.
• A negative z-score: Indicates the data point is below the mean.
• A z-score of zero: Means the data point's score is identical to the mean.

To calculate a z-score, you subtract the mean from the individual data point and then divide the result by the standard deviation. The formula is as follows: \[z = \frac{{X - \mu}}{{\sigma}}\]Where:

• \(X\) is the data point,
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

This calculation allows different types of data to be compared on the same scale which greatly simplifies probability calculations.

Calculating the probability using z-scores involves using the properties of the standard normal distribution. This probability represents the likelihood that a value falls between two points within the distribution.
To find the probability, we compare the area under the standard normal curve between the specified z-scores.
In the exercise at hand, we wanted to find the probability that the z-score is between -1.38 and 0. The steps included identifying the cumulative probabilities for each boundary and calculating the difference:

• First, find the probability that z is less than 0, which is 0.5 (as zero is the mean of the standard normal distribution).
• Next, look up the z-score -1.38 in the z-table to find its probability, approximately 0.0838. This step tells us the likelihood that a z-score is less than -1.38.
• Subtract 0.0838 from 0.5 to find the probability that z falls between -1.38 and 0, which is roughly 0.4162.

Ultimately, probability calculation using z-scores turns the z-scores into meaningful quantitative insights about where the data lie within a distribution.

The z-table, or standard normal table, is an essential tool in statistics that helps in determining the probability that a statistic is observed below, above, or between points on the standard normal distribution.
Each entry in the z-table corresponds to the area or probability under the standard normal curve to the left of a given z-score.
Here's how to effectively use the z-table:

• Find the row of the table that represents the integer and first decimal place of your z-score (e.g., for z = -1.38, look for -1.3).
• Look across the top (or the side, depending on your table format) to find the column that matches the second decimal place of your z-score (e.g., 0.08 for -1.38).
• The intersection of this row and column gives the cumulative probability for that z-score.

Using the z-table helps in understanding and applying the normal distribution to solve problems, such as finding the probability that a z-score falls within a specified range. This makes it a handy reference for many statistical analyses.