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A Z-score is a statistical measurement that describes how many standard deviations a data point is away from the mean of a data set. In the context of the standard normal distribution, which has a mean of 0 and a standard deviation of 1, a Z-score tells us precisely where a particular point falls within the distribution.

Calculating a Z-score is straightforward:

• Z = (X - μ) / σ

Where X is the data point, μ is the mean, and σ is the standard deviation. In a standard normal distribution, since μ is 0 and σ is 1, the formula simplifies.

• A positive Z-score indicates the data point is above the mean.
• A negative Z-score shows it is below the mean.

When we look for values greater than a certain Z-score, as in our problem, we determine the probability of observations that lie beyond that point in the distribution.

In statistics, probability measures the likelihood of a specific event occurring out of all possible outcomes.

For a standard normal distribution, probabilities are represented by the area under the curve. This model assumes that outcomes follow a bell pattern where most observations cluster around the mean, creating a symmetric curve.

Understanding probabilities in this distribution helps in:

• Estimating how likely a value will fall within a specific range.
• Making informed predictions and decisions based on data.

For example, if you seek the probability of observations greater than a Z-score of 1.78, you are looking at the proportion of the area to the right of this score in the standard normal distribution.

The Z-table, also known as the standard normal table, is a tool used to quickly find the cumulative probability associated with a particular Z-score. It lists the percentage of values to the left of a given Z-score in a standard normal distribution.

By looking at the Z-table:

• You can determine the likelihood of a score being less than a given value.
• The further you move away from 0, either positively or negatively, the lower the cumulative probability becomes.

When using the Z-table, locate the Z-score you are interested in. For 1.78, look up this value to see how much of the data lies below it. In our exercise, for a Z-score of 1.78, the cumulative probability is found to be about 0.9625.

Cumulative probability is the total probability that a random variable is less than or equal to a given value. In a standard normal distribution, it indicates what portion of the data falls under the curve to the left of a specified Z-score.

To find the area to the right (as the problem requires for Z > 1.78), subtract the cumulative probability from 1:

• P(Z > z) = 1 - P(Z < z)

This calculation gives the probability that a data point is above a certain threshold. For instance, if the cumulative probability for Z = 1.78 is 0.9625, the probability for Z > 1.78 is:

• P(Z > 1.78) = 1 - 0.9625 = 0.0375

This shows only a small fraction of values lie beyond 1.78 in the distribution, corresponding to the tail end of the curve.