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The z-score is a fundamental concept in statistics, especially when talking about normally distributed data. Simply put, a z-score tells us how far away a particular data point is from the mean in terms of standard deviations. By calculating a z-score, we can understand the position of the data point relative to the overall distribution.

To calculate a z-score, we use the formula:

• Where \( z = \frac{x - \mu}{\sigma} \)
• \( x \) is the data point of interest (e.g., 50 mg/l in our problem)
• \( \mu \) is the mean (e.g., 27 mg/l)
• \( \sigma \) is the standard deviation (e.g., 14 mg/l)

In our given problem, we find the z-score to be 1.64, meaning that 50 mg/l is 1.64 standard deviations above the mean of 27 mg/l.

This z-score helps us to determine probabilities associated with the normal distribution using z-tables. By converting data into z-scores, we can handle various types of normally distributed data with a standard method.

Probability in the context of normal distribution helps you determine the likelihood of a certain event happening under the normal curve. Here, it's crucial to understand that the normal curve represents a set of probabilities that total to 1.0.

After finding a z-score, we use it to determine the probability that a particular event differs from the average. This involves consulting a z-table, which maps each z-score to an area under the standard normal curve.

• For example, a z-score of 1.64 provides a probability of 0.9495, which represents the probability of a value being less than 1.64 standard deviations above the mean.
• To find the probability of the daily discharge exceeding 50 mg/l (\( P(Z > 1.64) \)), you subtract the table value from 1, since the z-table provides the probability for \( P(Z < 1.64) \).

Thus, \( P(Z > 1.64) = 1 - 0.9495 = 0.0505 \), indicating that there is a 5.05% probability for the discharge to exceed 50 mg/l.

This process shows how probability provides a way to quantify the chance of outcomes based on the normal distribution.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the case of the normal distribution, it determines how spread out the data values are around the mean.

A smaller standard deviation indicates that the data points tend to be close to the mean, whereas a larger standard deviation means the data points are spread out over a wider range.

• In our example, the standard deviation is 14 mg/l, which helps quantify the spread of the daily discharge levels around their mean value of 27 mg/l.
• A crucial part of understanding standard deviation is recognizing that it serves as a baseline unit of measurement when calculating z-scores.

Standard deviation is fundamental in assessing the reliability and predictability of data, enabling us to make informed judgments about how much variation we can expect from the mean. This understanding is essential for analyzing problems involving normally distributed data and determining likely outcomes.