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A z-score is a way to determine how far away a specific data point is from the mean of a standard normal distribution. It measures how many standard deviations a data point is from the mean value. The formula for calculating a z-score is: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Understanding z-scores is crucial because they allow us to compare data points from different normal distributions and determine their relative positions. - A positive z-score indicates that the data point is above the mean.- A negative z-score shows that it is below the mean.- A z-score of zero means the data point is exactly at the mean.By interpreting z-scores, we gain insights into how typical or atypical a particular observation is within a dataset.

Probability is a measure of how likely an event is to occur. In the context of normal distributions, we use probability to determine the likelihood of a z-score falling within a certain range. The total area under the standard normal curve is 1, representing 100% certainty that any data point will fall somewhere on the curve. For our example, we looked for the probability that the z-score is greater than or equal to -1.50. This is represented by the area under the curve to the right of this value. Calculating probabilities with z-scores involves finding the area under parts of the normal curve. To calculate the specific probability of an event: - Determine the relevant z-score. - Use tools like the z-table to find the area to the left or right of the z-score. - Subtract this value from 1 to find probabilities when needed. Understanding probability helps us predict and make decisions based on data, which is especially important in fields like statistics, economics, and everyday decision-making.

The z-table, also known as the standard normal table, is an essential tool for finding probabilities associated with z-scores in a standard normal distribution. This table shows the area to the left of a given z-score value. When dealing with probabilities, these areas correspond to the percentage of data that falls below a particular z-score. To use a z-table, follow these steps: - Locate the z-score's row based on the first decimal (e.g., -1.5 for a z-score of -1.50). - Identify the column based on the hundredth decimal point (e.g., .00 if the z-score is exactly -1.50). For a z-score of -1.50, the area is approximately 0.0668, as per the z-table. This means 6.68% of the data lies below this z-score in the distribution. To find the probability for values above the z-score: - Subtract the area given by the table from 1. Being proficient with a z-table allows us to quickly find how data is distributed across a normal curve, facilitating effective analysis and predictions.