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When we talk about the level of confidence in statistics, we are referring to the probability that a certain range, calculated from the sample data, contains the true population parameter.
For example, a 90% level of confidence means we are 90% certain that the true value lies within our interval.
This interval is determined by the critical value of the Z-score, which corresponds to our desired level of confidence.
The remaining 10% is distributed in the tails of the normal distribution—5% in the left tail and 5% in the right tail.
By understanding this concept, you can have a clear idea of the likelihood that your interval estimate actually contains the real population parameter.

The normal distribution, also known as the Gaussian distribution, is a symmetrical, bell-shaped distribution that is commonly used in statistics.
It is defined by its mean and standard deviation, and it has several key properties:

• The mean, median, and mode of a normal distribution are all equal.
• The curve is symmetric around the mean.
• About 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data is within two standard deviations.
• Nearly all the data (99.7%) lies within three standard deviations.

When dealing with a normal distribution, Z-scores can help determine the number of standard deviations a data point is from the mean.
This is useful in finding probabilities and critical values for statistical tests.

A Z-table, also known as the standard normal table, is a mathematical table used to find the cumulative probability associated with a particular Z-score in a standard normal distribution.
The Z-score represents the number of standard deviations a data point is from the mean.
To use a Z-table:

• Find the Z-score that corresponds to your area of interest. For example, for a 90% confidence level, we look for a Z-score with 95% cumulative probability.
• Locate the intersection of the row and column that corresponds to your Z-score. This will give you the cumulative probability up to that Z-score.
• For inverse lookup, match the cumulative probability in the table to find the Z-score.

In the example exercise, we needed to find the Z-score for the area to the left of 0.95, which was found to be approximately 1.645.
This Z-score tells us how far our critical value is from the mean in a standard normal distribution, allowing us to derive the critical values for specified levels of confidence.