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In statistics, the z-score is a measure that describes a value's position in relation to the mean of a dataset when measured in terms of standard deviations. This concept is essential in understanding how data points relate to the overall distribution. The formula to calculate the z-score is:

• \( z = \frac{X - \mu}{\sigma} \)

Here, \( X \) represents the value you are examining, \( \mu \) is the mean of the dataset, and \( \sigma \) is the standard deviation. The result gives the number of standard deviations \( X \) is away from the mean.
This measure allows us to standardize different data points, making it easier to compare them when they come from different places or scales. When applied to the airline baggage problem, it helps by converting the specific weight of 50 pounds into a z-score. In our case, the z-score of 1.5625 tells us that 50 pounds is 1.5625 standard deviations above the average baggage weight.

Standard deviation is a statistical measure that represents the spread or dispersion of a dataset. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation shows more spread. For our exercise, the standard deviation of baggage weights is 3.2 pounds. This means that most passengers' luggage will weigh within 3.2 pounds of the average weight of 45 pounds.
Standard deviation is crucial in calculating z-scores because it normalizes the distribution. By knowing how spread out the data is, you can determine how unusual or typical a specific data value is. For instance, knowing that 50 pounds is outside the standard deviation range allows us to say it's common or uncommon relative to typical weights.

Probability in this context refers to the likelihood of a particular outcome occurring, such as the weight of the baggage exceeding a certain value. When using normal distributions, probability is determined by the area under the curve in a graph, which depicts different possible outcomes. Normal distributions are symmetrical and bell-shaped, making them predictable based on mean and standard deviation.
Using the z-score calculated from our example, we can find the probability of an event. The z-table helps translate this z-score into a probability that represents the percentage of data under a curve. This helps in our baggage scenario to determine how many passengers will incur the excess weight fee by showing the weight distribution relative to exceeding 50 pounds.

The z-table, or standard normal distribution table, is a mathematical tool that shows the percentage of values to the left of a given z-score on a standard normal distribution. It helps convert z-scores into probabilities, making it easier to understand how often real-world data fall within certain ranges.

• It's important when you need to find a percentage or proportion of data that lies below or above a particular z-score value.

To use the z-table effectively, once you have your z-score, you look it up in the table, which acts as a lookup chart.
For the airline baggage problem, the z-score of 1.5625 shows us that approximately 94.06% of baggage weights are below 50 pounds. This leads to the final calculation where about 5.94% of the passengers' baggage exceeds that weight, thus incurring a fee.