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The Z-score is a fundamental concept in statistics, especially when dealing with normal distributions. It is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a data point's Z-score is 0, it indicates that the data point's score is identical to the mean score. The formula to calculate the Z-score is:\[Z = \frac{X - \mu}{\sigma}\]where:

• \(X\) is the value to be standardized
• \(\mu\) is the mean of the distribution
• \(\sigma\) is the standard deviation

The Z-score tells us how many standard deviations away from the mean a particular data point is. For example, a Z-score of 1.0923 means the value is 1.0923 standard deviations above the mean. This makes it easier to determine the proportion of data points above or below a certain value in the context of a normal distribution. The Z-score is crucial because it helps translate a raw score into a location on the standard normal distribution, which can then be used to find probabilities.

Standard deviation is a statistical measurement that quantifies the amount of variation or dispersion in a set of data values. It gives insight into how much the individual data points in a dataset differ from the mean (average) value of the dataset. A small standard deviation indicates that the data points tend to be close to the mean, while a large standard deviation indicates that the data points are spread out over a wider range. The formula for standard deviation is:\[\sigma = \sqrt{\frac{\sum{(X_i - \mu)^2}}{N}}\]where:

• \(X_i\) represents each data point
• \(\mu\) is the mean of the data
• \(N\) is the number of data points

In the context of a normal distribution, standard deviation plays a pivotal role. It helps determine the "spread" of the bell curve and thus significantly impacts the shape of the distribution. In our example, the standard deviation helps us understand how the tax refunds are distributed around the mean of $2,290. It also aids in calculating Z-scores, which further guides us in determining cumulative probabilities associated with certain refund amounts.

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specified number. In the context of statistics and the normal distribution, it signifies the area under the probability distribution curve to the left of a given value. To find cumulative probabilities, we rely on the standard normal distribution table, often known as the Z-table. Every Z-score corresponds to a cumulative probability, showing the proportion of values lying below it. For a Z-score of 1.0923, for instance, the cumulative probability is approximately 0.8621. This means about 86.21% of the data lies below this value. Once we have the cumulative probabilities, we can easily find:

• Percentages more or less than a certain Z-score by subtracting the cumulative probability from 1 (or 100% for percentages).
• Probabilities between two data points by calculating the difference between their cumulative probabilities.

Employing these techniques allows us to solve real-world problems, such as determining what percentage of tax refunds fall beyond specific cut-off points or within a particular range, as demonstrated in the exercise.