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Understanding the concept of a Z-score is fundamental in statistics, especially when dealing with normal distributions. A Z-score, also known as a standard score, is a numerical measurement used in statistics to describe a value's relationship to the mean of a group of values. It's calculated using the formula:
\[ Z = \frac{(X - \text{mean})}{\text{standard deviation}} \]
In the context of our gasoline tank problem, the mean (\(\text{mean}\)) is 15 gallons, and the standard deviation (\(\text{standard deviation}\)) is 0.1 gallon. To find the Z-score for a tank that holds at most 14.8 gallons, we substitute the actual capacity (\(X\)) with 14.8 and perform the calculation. The Z-score then represents how many standard deviations away from the mean our actual capacity is.
A negative Z-score means the actual capacity is below the mean, while a positive Z-score means the capacity is above the mean. Through Z-scores, we can compare different values on a standard scale, even if the original data points have different units or scales.

The normal distribution, which is symmetric and bell-shaped, represents the distribution of various data across many fields. An important feature of the normal distribution is that the total area under the curve is equal to 1, or 100%, which embodies all possible outcomes.
When we talk about the 'area under the normal curve', we are referring to the probability of a random variable falling within a particular range. This region is found using the calculated Z-scores, which correspond to specific probabilities according to the standard normal distribution table or can be computed using a calculator with statistical functions.
In the gasoline tank exercise, after calculating the Z-score for 14.8 gallons, we can use the standard normal distribution table to find the area to the left of this Z-score, representing the probability of selecting a tank with at most 14.8 gallons. The result is the likelihood that a randomly selected tank from the population will meet this specific condition.

Probability for a range of values in a normal distribution involves finding the likelihood that a variable falls between two different points. For instance, with our gasoline tank example, to determine the probability that a tank will hold between 14.7 and 15.1 gallons, we calculate the Z-scores for each gallon measurement and then find the areas corresponding to these Z-scores.
You can think of it as shading the area under the normal curve between two Z-scores. The probability is the amount of that shaded region. The area for a single point on the curve is zero since the curve is continuous and has infinitely many points. That's why we look at ranges—it's the way we can measure an event's likelihood in continuous distributions.
For problems involving multiple independent events—like selecting two tanks and wanting them both to hold at most 15 gallons—we calculate the probability of one event and then use the rules of multiplication for independent events to find the combined probability. The independence of the tanks' capacities means that the filling of one tank does not influence the other, which is a key assumption for our calculations to be correct.