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Understanding z-scores is crucial for interpreting data in relation to a normal distribution. A z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. The formula for calculating a z-score is given by:
\[ Z = \frac{ (X - \mu) }{ \sigma } \]
where \( X \) represents the value in question, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation. In practical application, such as the temperature study, with a body temperature of \( 98.6^\circ F \), a mean temperature of \( 98.4^\circ F \), and a standard deviation of \( 0.70^\circ F \), the z-score would be calculated using the provided data:
\[ Z = \frac{ (98.6 - 98.4) }{ 0.70 } \]
This value then helps in determining how far away the observed temperature is from what's typical within the group of healthy women. A positive z-score indicates a value above the mean, while a negative z-score signifies a value below the mean.

Significance of Z-scores

Z-scores are not only important in indicating the position of an individual data point but are also used in various statistical tests and confidence intervals. They are the backbone for understanding the probability of occurrences within a standard normal distribution.

When working with normally distributed data, the concept of probability and percentages is pivotal. Probability, in the context of a normal distribution, expresses the likelihood of a random variable falling within a particular range. Meanwhile, percentages help us convey this probability in a more intuitive way.

After calculating the z-score, the next step often involves using a z-table (also referred to as a standard normal distribution table) to find the corresponding probability. This table shows the probability of a random variable falling between the mean and any z-score within a normal distribution. To convert this to a percentage, you can simply multiply the probability by 100.

For example, if you find a probability of 0.75 from the z-table for a particular z-score, this translates to a 75% chance that the randomly selected value lies between the mean and that z-score. Deciphering such tables and converting probabilities to percentages enables statisticians and data analysts to make predictions and decisions based on the distribution of the data.

Practical Application

In exercise (a) from the original problem, the z-score helps determine the percentage of women with temperatures below \( 98.6^\circ F \). By referring to the z-table with the calculated z-score, we obtain the percentage which represents the probability of finding a woman with a lower temperature—a foundational concept for statistical analysis in various fields.

Percentiles are integral to the field of statistics, offering a way to understand and describe the relative standing of a value within a data set. By definition, the k-th percentile is a value below which k percent of observations in a given distribution fall. This concept allows us to compare individual scores to a broader data set.

In a normal distribution, each percentile corresponds to a unique z-score. For instance, the median (50th percentile) has a z-score of 0 because it's precisely at the mean. To find the z-score associated with any other percentile, you can use statistical tables or software designed for this purpose.

Once you have this z-score, you can convert it into the actual value you are looking for by manipulating the z-score formula as mentioned in the original problem's step 4:
\[ X = Z \cdot \sigma + \mu \]
For example, to find the temperature corresponding to the 76th percentile in a given dataset, we start with the z-score associated with that percentile. Using the mean and standard deviation of the data, we can find the exact temperature value. This enables us to say, with a certain degree of confidence, that 76% of temperatures in our data set are below this value, and 24% are above. This is a powerful statistical tool used across research, business, and healthcare industries to make informed decisions based on the distribution of data.