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The Z-score is a statistical measure that tells you how far a particular data point is from the mean, measured in terms of standard deviations. It’s a way to standardize a number relative to its dataset. For instance, if you have a Z-score of 1.0, this means that your data point is exactly one standard deviation away from the mean of the distribution. Here’s how you calculate it:

• Use the formula: \( Z = \frac{X - \mu}{\sigma} \).
• \( X \) is the value you are examining.
• \( \mu \) is the mean of the data set.
• \( \sigma \) is the standard deviation.

In the context of the AA battery example, to find out how unusual a voltage of 1.54V is in comparison to the average voltage, we apply the Z-score formula as follows: \( Z = \frac{1.54 - 1.50}{0.05} = 0.8 \).
A Z-score of 0.8 means that the battery's voltage is 0.8 standard deviations higher than the average voltage. Each Z-score corresponds to a position in the standard normal distribution, which helps us understand the likelihood or rarity of the value.

Mean, often symbolized as \( \mu \), is the average value of a dataset. It provides a central value for the data. For the AA batteries, the mean voltage is given as 1.50V, which indicates the average voltage level for new batches of batteries.
The standard deviation, denoted by \( \sigma \), measures the spread or the variability of the dataset around the mean. In our example, \( \sigma = 0.05 \) V. Here’s how to interpret these terms:

• **Mean (\( \mu \))** is the sum of all data points divided by the number of data points. It helps to locate the center of the dataset.
• **Standard Deviation (\( \sigma \))** indicates how much individual data points deviate from the mean.

For the batteries, the standard deviation of 0.05V suggests that most of the voltages are close to the mean, 1.50V. Understandably, a smaller standard deviation implies that the voltages don’t spread too widely from the mean, reflecting in the calculated Z-score when assessing unusual voltage levels.

The Z-table, or standard normal distribution table, is a useful tool that helps you find probabilities associated with a specific Z-score. It provides the cumulative probability of a value occurring less than the given Z-score in a standard normal distribution.
Here’s how it works:

• A Z-table provides areas under the curve to the left of a Z-score value.
• For a Z-score of 0.8, the Z-table reads about 0.7881, indicating 78.81% of data falls below this score.

In the case of the AA batteries, when looking up a Z-score of 0.8, the Z-table reveals that about 78.81% of batteries have a voltage less than 1.54V. This signifies that a little over three-fourths of the batteries have voltages below 1.54V. Using the Z-table is crucial in transforming Z-scores into practical percentages, enabling clearer interpretation of statistical data. Within the realm of normal distribution, the Z-table is indispensable for probability calculations, where direct interpretations of standard deviations just won't suffice alone.