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A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. In simpler terms, it tells you how many standard deviations a particular value is away from the mean.

For example, if the mean number of hours worked per week by medical residents is 81.7 and a specific resident worked 90 hours, a Z-score will help you determine how relatively far this 90-hour work week is from the average.

To calculate a Z-score, you use the formula:
\( Z = \frac{X - \bar{X}}{\text{standard deviation}} \) where:

• \bar{X}\bar{X}: mean of the data set, in this case, 81.7 hours
• X: specific value from the data set, in this case, the number of hours worked
• standard deviation: how spread out the numbers are from the mean, in this case, 6.9

For the 75th percentile, we find a Z-score of approximately 0.674. This means that the value at the 75th percentile is 0.674 standard deviations above the mean.

A normal distribution, also known as a bell curve, is a type of continuous probability distribution for a real-valued random variable. It is symmetric and describes how the values of a variable are distributed.

In a normal distribution:

• Most of the data points tend to cluster around the mean, creating the peak of the graph at the center.
• The probabilities for values further from the mean taper off equally in both directions, creating symmetrical tails.
• The mean, median, and mode of the distribution are all equal.

Understanding the shape and properties of a normal distribution is crucial when working with Z-scores and percentiles, as these concepts rely on the data being normally distributed.

In the problem above, we assume that the number of hours worked per week by medical residents follows a normal distribution with a mean of 81.7 and a standard deviation of 6.9. This assumption allows us to use Z-scores and percentile calculations accurately.

Percentiles help you understand the relative standing of a value within a data set. A percentile indicates the value below which a given percentage of observations fall. For instance, the 75th percentile is the value below which 75% of the observations in a dataset fall.

To find a specific percentile in a normally distributed dataset, you use the corresponding Z-score, which you can find in a Z-table. You then use the mean and standard deviation of the dataset to calculate the actual value, using the formula: \( X = \bar{X} + Z \times \text{standard deviation} \)

For example, to find the 75th percentile, we use the Z-score of 0.674,
then calculate: \( X = 81.7 + 0.674 \times 6.9 \). This results in approximately 86.35 hours.

Percentiles are essential for understanding how an individual data point compares to the rest of the dataset. In the case of the medical residents, knowing the 75th percentile helps you determine that 75% of residents work fewer than 86.35 hours per week.