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The mean, often referred to as the average, is a key concept in statistics. It represents the central point of a dataset, especially when the data follows a normal distribution.
In our case, we're dealing with the sales of romance novels, which are more than 470,000 in 40% of cases and exceed 500,000 in 10% of cases. This information allows us to calculate the mean sales value.

The process begins with setting up an equation using the formula for the normal distribution: \[ X = \mu + Z\sigma \]- "\(\mu\)" is the mean, the value we aim to find.- "\(Z\)" is the z-score associated with a particular cumulative probability.- "\(\sigma\)" is the standard deviation.

Using the z-scores corresponding to the given probabilities (0.253 for 470,000 and 1.282 for 500,000), we set up two equations:- \( 470,000 = \mu + 0.253\sigma \)- \( 500,000 = \mu + 1.282\sigma \)
By solving these equations, we find that the mean, \(\mu\), is approximately 462,624. This tells us that, on average, the annual sales of the novels hover around this value.

Standard deviation is a crucial statistic that measures the amount of variation or dispersion in a set of values. When data points are spread out over a wider range of values, the standard deviation is larger; when they are closer to the mean, the standard deviation is smaller.
In the context of our exercise, once we identified the z-scores corresponding to our cumulative probabilities, we were able to set up two equations.

The difference between the two information points, from sales of 470,000 and 500,000, was used to determine the standard deviation:- We know from our equations that \[ 30,000 = 1.029\sigma \]By solving this, we find \[ \sigma = \frac{30,000}{1.029} \approx 29,148 \]
This indicates that the sales figures tend to vary by about 29,148 from the mean. This variability informs retail expectations, helping them manage stock and predict future sales trends effectively.

A z-score, also known as a standard score, represents how many standard deviations an element is from the mean. Z-scores are used in statistics to understand where a specific data point lies in relation to the normal distribution of a dataset.
When looking at the sales of romance novels, a 60% cumulative probability translates to a z-score of approximately 0.253, meaning those sales are just slightly above the mean. Similarly, a 90% cumulative probability translates to a z-score of approximately 1.282, indicating much higher sales relative to the overall distribution.

The formula to find a z-score is:\[ Z = \frac{X - \mu}{\sigma} \]Where:- \(X\) is the value of the data point.- "\(\mu\)" is the mean.- "\(\sigma\)" is the standard deviation.
Using lookup tables or tools, we can translate cumulative probabilities to z-scores, which in turn help solve for unknowns like mean and standard deviation, as we did in this problem. Understanding z-scores is essential in statistics for assessing probabilities of observing certain data points within a distribution.