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A normal distribution is a common way to describe how a set of values are spread out, especially when dealing with natural phenomena or measurement data. This distribution is symmetrical, which means it has a bell curve shape where most of the data points cluster around the mean. The two main parameters defining a normal distribution are the mean (average of all data points) and the standard deviation (which we will discuss later in more detail). The mean is at the center of the curve, and the standard deviation determines the width of the bell curve. In the context of this exercise, we assume that the annual costs to attend a private university follow a normal distribution. This means most students pay amounts close to the average of $22,218, with fewer students paying much less or much more than this amount. Each specific € value of the normal distribution provides vital info about how costs distribute around the average in a real-world setting.

Percentiles are a great way to understand the position of a data point within a larger dataset. The 95th percentile is particularly significant because it represents the value below which 95% of the observed data fall. When we say that 95% of students pay an amount less than a certain value, we're referencing the 95th percentile cost in the specific context of university fees. To find this specific value, we utilize the z-score, which is a measure of how many standard deviations an element is from the mean. For the normal distribution, the z-score for the 95th percentile is around 1.645. Knowing this helps us understand how far from the mean this cost is actually found, providing us with valuable insight into the upper range of tuition fees.

Standard deviation is a prominent statistical parameter that indicates the amount of variation or dispersion in a set of data values. A low standard deviation means the values are clustered close to the mean, while a high standard deviation indicates a wide range of values. In the context of the university costs, the standard deviation is $4,500. This shows that most students' annual costs are within this financial scope from the average of $22,218. This measurement is key to determining the 95th percentile value, as the standard deviation is multiplied by the z-score to find out how much one should add to the mean to reach that specific percentile. This makes it a pivotal aspect in grasping the concept of variation within the annual costs.

The mean, or average, is at the heart of many statistical analyses as it represents the central value of a dataset. Calculating the mean involves adding all the data values together and dividing by the number of data points. In this problem, the mean of $22,218 indicates the average annual cost for students attending a private university. Furthermore, it serves as the center of our normal distribution, around which the majority of data points will naturally cluster. Understanding the mean is essential as it is the starting point for calculating any percentile value within a normal distribution. By knowing the mean and then adjusting by standard deviations (using the z-score for specificity), we can determine the costs at various percentiles, giving us a clearer picture of the cost distribution landscape.