91Ó°ÊÓ

The z-score is a fundamental concept in statistics, especially when dealing with normal distributions. It helps us understand how far and in what direction a data point is from the mean of the data set. For any given data point, the z-score indicates how many standard deviations it is away from the mean. This allows us to compare different data points within a distribution. To calculate a z-score, we use the formula:\[ z = \frac{X - \mu}{\sigma} \]where:

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

By standardizing data points into z-scores, we can easily compare and interpret datums based on their position in the distribution. For example, if a cruise had 1,970 passengers with a z-score of 1.25, it indicates that it's 1.25 standard deviations above the mean of 1,820 passengers.

The mean and standard deviation are key statistical measures that summarize a data set's characteristics. The mean, often referred to as the average, is the central value of a data set. It is calculated by adding all the individual data points and then dividing the sum by the number of data points. For the Carnival Sensation cruises, the mean number of passengers is 1,820, representing the central tendency of passenger distribution across cruises. On the other hand, standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation means that the data points tend to be close to the mean, whereas a high standard deviation indicates wider spread around the mean. In the context of the cruise problem, the standard deviation is 120. This number tells us that the number of passengers typically varies by about 120 from the mean of 1,820. Knowing the standard deviation helps us understand how typical or unusual a particular passenger count is on a cruise.

Percentile calculation is a statistical method used to understand the relative standing of a value within a data set. A percentile indicates the value below which a given percentage of observations in a data set fall. For example, if a cruise passenger count falls at the 90th percentile, it means that 90% of cruises have fewer passengers, and only 10% have more. This concept is essential for understanding distributions and identifying thresholds in data sets.To calculate the number of passengers in the lowest 25% of cruises, we first determine the z-score that corresponds to the 25th percentile. In the context of a normal distribution, this z-score is approximately -0.675. We can then plug this value into the z-score formula to solve for the actual number of passengers:\[-0.675 = \frac{X - 1,820}{120}\]Rearranging and solving for \(X\), we find:\[X \approx 1,738\]Thus, cruises with fewer than 1,738 passengers represent the bottom 25% of the distribution. This calculation is crucial for assessing how different data points rank within the overall data set.