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The z-score is a statistical measurement that helps us understand how far away a specific data point is from the mean of a data set. It is expressed as the number of standard deviations a data point is from the mean.
To calculate the z-score, you use the formula: \[ z = \frac{x - \mu}{\sigma} \]where:

• \(x\) is the value you are examining
• \(\mu\) is the mean of the data
• \(\sigma\) is the standard deviation

In the exercise, z-scores were calculated to find the probability of brain weights between specific limits. For example, a z-score of -0.75 means the value is 0.75 standard deviations below the mean. This helps us locate the position of the value in the normal distribution curve.

The mean and standard deviation are key concepts in statistics, especially when dealing with normal distributions. The mean represents the average of all data points, providing a central reference point for data distribution. On the other hand, the standard deviation measures the amount of variability or dispersion in a data set.
For instance, a mean (\(\mu\)) of 1400 g for the brain weights suggests that this is the center of the data set. The standard deviation (\(\sigma\)) of 100 g signifies that most of the brain weights are within 100 g above or below the mean. This concept is crucial for understanding how spread out the data is, and allows the computation of z-scores necessary for probability calculations.

Calculating probabilities in a normal distribution involves using z-scores and a z-table (or standard normal distribution table). Once we convert our data points to z-scores, we can look these up in a z-table to find the corresponding probabilities.
In part (a) of the exercise, finding the probability that brain weights are between 1325 g and 1450 g required calculating the z-scores for these limits. After obtaining the z-scores, we found their probabilities using the z-table and subtracted them to find the probability for the range. This probability tells us how likely it is to find brain weights between these given values, expressed as a percentage of the total distribution.

Determining the percentage of a population within certain ranges in a normal distribution involves interpreting probabilities. By converting specific data points to z-scores and using a z-table, we can estimate what percentage of the population falls within a certain range.
For example, the exercise calculated that 46.49% of the population have brain weights between 1325 g and 1450 g. For part (b), the calculation showed that approximately 21.19% of the population has brain weights exceeding 1480 g. These percentages help us understand how groups are distributed within the entire population, providing a clearer picture of the data distribution pattern.