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The Z-score is an essential concept when dealing with standard normal distributions. It helps us understand how a single data point relates to the rest of the dataset. The Z-score measures the number of standard deviations an observation is from the mean. In the context of a standard normal distribution, where the mean is zero (\(\mu = 0\)) and the standard deviation is one (\(\sigma = 1\)), the Z-score directly represents our observation's position along this distribution.

When you have a Z-score of 1.78, like in our exercise, it indicates that the value is 1.78 standard deviations away from the mean. This score helps in determining the likelihood of occurrence of this or more extreme values, which is a core aspect of probability calculations in statistics. Understanding Z-scores paves the way for systematic approaches to various statistical analyses, such as hypothesis testing and confidence intervals.

In statistics, one of the key applications of the Z-score is in probability calculation. We calculate the probability to understand the likelihood of a certain event or observation occurring. For our current problem, the task was to calculate the probability of observing a value greater than 1.78 in a standard normal distribution.

Here's how this works:

• First, you find the probability of a Z-score less than 1.78. This gives us the cumulative probability for all values up to 1.78 in the distribution.
• Once we have this cumulative probability, calculating the probability for the complementary event (values greater than 1.78) is simple. Since total probability equals 1, you subtract the cumulative probability from 1.

Therefore, the calculation goes like this: if \(P(Z < 1.78) = 0.9625\), then \(P(Z > 1.78) = 1 - 0.9625 = 0.0375\). This step-by-step process ensures accuracy and helps compare the result to predefined answers, confirming the correct probability estimation.

The Z-table is a fundamental tool in statistics for looking up probabilities associated with the standard normal distribution. Think of it as a reference guide existing to simplify our calculations when dealing with Z-scores.

To use a Z-table effectively, follow these steps:

• Identify the Z-score you need to evaluate. In our problem, this was 1.78.
• Locate this Z-score in the Z-table. The table is structured with Z-scores and their corresponding cumulative probabilities.
• Read off the cumulative probability. In our example, for a Z-score of 1.78, the table shows approximately 0.9625, which is the probability that a value is less than 1.78.

Once you have this cumulative probability, you can use it to find probabilities for values greater than your Z-score by subtracting from 1. This makes the Z-table not just a helpful tool, but an indispensable resource for simplifying probability calculations in statistics.