91Ó°ÊÓ

The concept of a z-score is key in understanding normal distribution. A z-score helps us determine how far away a particular value is from the mean, measured in standard deviations. This is extremely useful to compare different data points within a normal distribution. To calculate the z-score, use the formula: \[ Z = \frac{X - \mu}{\sigma} \] Where:

• \(X\) is the value you are evaluating.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For example, if you want to find out how different a customer's waiting time of 3 minutes is from the average (8 minutes in this exercise), you substitute into the formula: \[ Z = \frac{3 - 8}{2} = -2.5 \] A z-score of -2.5 indicates that 3 minutes is 2.5 standard deviations below the mean.

Understanding probability in the context of normal distribution means predicting the likelihood of a certain outcome. The normal distribution, which often appears as a bell-shaped curve, enables us to use z-scores to find probabilities for different outcomes. Once you have a z-score, you can use a z-table (or standard normal distribution table) to find the probability. The table tells you the probability of a value being less than the given z-score. For example, a z-score of -2.5 corresponds to a probability of 0.0062, meaning there's a 0.62% chance that a randomly selected customer will wait less than 3 minutes. Similarly, to find the probability of waiting times falling within a range, we can look up each z-score in the table and subtract the smaller probability from the larger one. For instance, the probability of waiting 10 to 13 minutes is found by subtracting the z-score probabilities of 10 minutes (0.8413) from 13 minutes (0.9938), resulting in 15.25%.

Standard deviation is a critical component of understanding the spread in a set of data. In the setting of a normal distribution, it denotes how much variation or "dipersion" exists from the mean. A smaller standard deviation means the data points tend to be close to the mean, while a larger standard deviation indicates more spread out data points. In the exercise scenario, the standard deviation of customer waiting times is 2 minutes. This gives us insight into how varied or consistent these times are. When calculating z-scores, the standard deviation is used to scale differences from the mean, making them unitless, which allows comparing them within the same data set. Thus, it aids in determining the likelihood of various waiting periods and how they diverge or converge around the average of 8 minutes. Understanding this spread helps the bank management anticipate and provide efficient customer service by minimizing extreme waiting times.

The mean is commonly known as the average, and it's a central value around which a normal distribution is organized. It's calculated by summing up all values and dividing by the number of values. In the exercise, the mean waiting time is 8 minutes, representing the central tendency of customer waiting times at the bank. This mean serves as a reference point for calculating z-scores, which explain how each waiting time compares to the average. It's also essential in understanding the symmetrical property of normal distribution, where about 68% of values lie within one standard deviation, 95% within two, and 99.7% within three standard deviations of the mean. A normal distribution with a well-understood mean helps predict the probability of different waiting times, guiding the branch manager's efforts in improving service efficiency. Understanding the mean and its role in normal distributions enables proactive adjustments to improve customer satisfaction in terms of waiting time.