91Ó°ÊÓ

The Z-score is a fundamental concept when working with normal distributions. It allows us to understand how far a specific value is from the mean of the distribution in terms of standard deviations. To compute the Z-score, we use the formula:

• \[ Z = \frac{X - \mu}{\sigma} \]

Here, \( X \) is the value for which we want to find the Z-score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

The Z-score tells us:

• How many standard deviations a particular value is from the mean.
• If the Z-score is positive, the value is above the mean; if negative, it is below.

In the context of the IRS tax refund example, calculating the Z-score for amounts like \( \$3,100 \) helps determine the position of these amounts within the normal distribution of all tax refunds.

Standard deviation is a key measure in statistics that describes the amount of variation or dispersion in a set of values.
It tells us how much the individual data points typically deviate from the mean of the data set.
A smaller standard deviation indicates that the data points are close to the mean, while a larger one suggests more spread out data.

In a normal distribution:

• About 68% of data falls within one standard deviation of the mean.
• Approximately 95% is within two standard deviations.
• Nearly all (99.7%) lies within three standard deviations.

In our example with tax refunds, the standard deviation is set at \( \\(450\).
This gives us insight into how much individual refunds typically vary from the average refund of \( \\)2,800\).
Understanding standard deviation helps us better grasp the overall spread and reliability of the values within a dataset.

Probability calculations using the normal distribution involve finding the likelihood of a particular range of values occurring.
This is where Z-scores and standard deviation play a crucial role.
By calculating Z-scores, we transform specific data values into a standard form, making it easier to find probabilities using a standard normal distribution table or calculator.

To find probabilities:

• First, calculate the Z-score of the values of interest.
• Next, use the Z-score to find the corresponding probability from a Z-table.
• Interpret the result: for values greater than a certain point, subtract the Z-score probability from 1.

In the refund problem, we determined the probabilities of refunds being over \( \\(3,100 \), between \( \\)3,100 \) and \( \\(3,500 \), and between \( \\)2,250 \) and \( \$3,500 \).
These calculations help in making informed predictions about the distribution of tax refunds. It shows how likely different refund amounts are and supports decision-making processes.