91Ó°ÊÓ

A z-score is a crucial concept when dealing with normal distributions. It essentially tells us how many standard deviations away a particular data point is from the mean. To calculate a z-score, use the formula: \[ z = \frac{X - \mu}{\sigma} \]Where:

• \(X\) is the observed value
• \(\mu\) is the mean of the data set
• \(\sigma\) is the standard deviation

A z-score of 0 means the data point is exactly at the mean. Positive z-scores represent values above the mean, while negative z-scores represent values below the mean. By converting data into z-scores, we can look up probabilities in a standard normal distribution table, enabling us to determine how common or rare a data point is within a distribution.
For example, in our exercise, converting 5 minutes to a z-score tells us how much longer than average these calls last compared to a typical call.

Standard deviation is a measure of how spread out the numbers in a data set are. In simpler terms, it tells us how much the values in a set typically differ from the mean (average). A smaller standard deviation means the values are closer to the mean, while a larger one indicates a wider spread.The formula for standard deviation is:\[ \sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}} \]Where:

• \(X\) represents each value in the dataset
• \(\mu\) is the mean
• \(N\) is the number of data points

In the context of our problem, knowing the standard deviation (0.60 minutes, in this case) allows us to calculate z-scores, which are used for probability calculations and understanding the distribution of phone call durations.

Probability calculations are fundamental for interpreting normal distributions. These calculations help us determine the likelihood of a data point falling within a certain range. Once data points are converted into z-scores, these can be used alongside z-tables or calculators to find probabilities. For instance, when you know the z-score related to a particular phone call duration, you can look it up in a z-table to determine the probability of such a call lasting that long. In our exercise, we calculated that the probability of a call lasting between 5 and 6 minutes is approximately 0.0905, or around 9.05%, by looking at the cumulative probabilities for the calculated z-scores. This assists in making informed decisions based on past data, which is particularly beneficial for business and statistical analysis.

Percentiles indicate the relative standing of a value within a data set. If you know a value's percentile rank, you understand what proportion of the data is below it. Percentiles are especially common in educational testing and statistical report presentations.To find percentiles in a normal distribution, you often use z-scores. For example, if you're interested in the value below which 96% of phone call durations fall, you find this by identifying the z-score that corresponds to the 96th percentile. In the exercise provided, we wanted the maximum duration for the longest 4% of calls, which means we looked for the 96th percentile. By using the z-score for this percentile, which is approximately 1.75, we could calculate the corresponding call duration using the formula:\[ X = \mu + z \cdot \sigma \]Understanding percentiles helps in setting benchmarks and interpreting relative performance across different datasets.