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Calculating a Z-score is a fundamental step when working with normal distributions. It helps you understand how far away a specific value is from the mean, in terms of standard deviations.
To calculate the Z-score, use the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

Here:

• \(X\) is the value in question, such as the tank capacity of 14.8 gallons.
• \(\mu\) is the mean, the average capacity, given as 15.0 gallons.
• \(\sigma\) is the standard deviation, which measures the spread, specified as 0.1 gallon.

For example, to find the Z-score for a tank holding 14.8 gallons:

• Substitute into the formula: \( Z = \frac{14.8 - 15}{0.1} = -2\).

This calculation shows that 14.8 gallons is 2 standard deviations below the mean.

Once you've calculated the Z-score, the next step involves determining the probability. This involves looking up the Z-score in the standard normal distribution table, or alternatively using statistical software. This table tells you the probability that a value is less than or equal to the given Z-score.
For instance, with a Z-score of -2:

• The probability corresponds to 0.02275.

This indicates that there is a 2.275% chance that a randomly selected tank will hold at most 14.8 gallons.
To find the probability that a tank holds between two values (e.g., between 14.7 and 15.1 gallons), calculate the Z-scores for both values.
The difference between their respective probabilities from the standard normal distribution provides the probability of seeing a value within that range.
Hence, the difference between probabilities for Z-scores \(Z1 = -3\) and \(Z2 = 1\) results in a probability of 0.83999, or about 83.999%.

The standard normal distribution is an essential concept in statistics. It's a normal distribution with a mean of 0 and a standard deviation of 1. Every normal distribution, regardless of its mean or standard deviation, can be converted into the standard normal distribution using the Z-score.
This allows for easier calculation of probabilities.

• It helps us determine how likely a particular outcome is.
• It simplifies comparing different data sets or variables since they can be analyzed on the same scale.

When working with Z-scores, probabilities can be quickly deduced by referencing the standard normal distribution table.
The conversion of any normal distribution to a standard form eliminates the variability associated with different scales, offering a standardized method to approach problems in statistics.
Applying it to our exercise, any tank's capacity can be transformed into a Z-score, which can then be used to determine probabilities effortlessly using the standard normal distribution.