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The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is a way to determine how many standard deviations an element is from the mean. To calculate the Z-score, use the formula: \[ Z = \frac{(X - \mu)}{\sigma} \] Here, \( X \) is the value from the dataset, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. The Z-score helps us understand the likelihood of a given temperature occurring under normal distribution. It effectively standardizes raw scores, enabling the comparison between different data sets or distributions.

• For example, if we have a body temperature of \( 98.6^{\circ} \mathrm{F} \), mean \( \mu \) of \( 98.1^{\circ} \mathrm{F} \), and standard deviation \( \sigma \) of \( 0.70^{\circ} \mathrm{F} \). You plug these into the formula and calculate the Z-score: \[ Z = \frac{(98.6 - 98.1)}{0.70} \]

This Z-score value can then be used to find the percentile or probability of this temperature within the normal distribution.

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. In the context of standardized tests or normal distributions, it tells us about the rank of a particular score. If a score is at the 76th percentile, it means that 76 percent of all other scores in the distribution are lower. Using the Z-score and the Z-table are essential steps in finding percentiles:

• Look up the cumulative probability corresponding to a Z-score in the Z-table. This gives you the percentile rank of a certain score.
• For example, to find out what temperature corresponds to the 76th percentile, we look up the Z-score for 0.76 (cumulative probability) in a Z-table. This process shows how our value stands in comparison to others in our dataset.

By understanding percentiles, we can effectively analyze data sets by determining how a specific value compares to the distribution as a whole.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means the data points tend to be close to the mean of the data set, while a high standard deviation means the data points are spread out over a wider range of values. In the context of normal distributions, the standard deviation helps determine the spread of data around the mean. It is crucial for calculating Z-scores and percentiles as it reflects the expected variability of data values:

• Standard deviation \( \sigma \) can be calculated from the data points but is often provided, as in our exercise, where \( \sigma = 0.70^{\circ} \mathrm{F} \).

The standard deviation is what transforms raw scores into Z-scores via the Z-score formula. Understanding how the standard deviation affects the normal distribution allows for deeper insight into analyzing different probabilistic scenarios or evaluating the variability observed in the dataset.