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In statistics, the **mean** is the average of a set of values. It is calculated by summing all the numbers and then dividing by the count of numbers. In our exercise, the mean amount that families spend on back-to-college items is given as \( \\(616.13 \). This value represents the center of our data distribution.
The **standard deviation**, on the other hand, measures how spread out the values are from the mean. It shows the average distance between each data point and the mean. A higher standard deviation indicates more variability in spending, while a lower one indicates that spending amounts are closer to the mean. In the given problem, the standard deviation is \( \\)120 \). This tells us the typical amount by which the spending of families might deviate from the mean amount of \( \$616.13 \).
Understanding these two parameters is essential because they define the normal distribution curve, which helps us compute probabilities of different spending scenarios.

A **Z-score** is a way to compare individual data points to a normal distribution. It represents the number of standard deviations a datapoint is from the mean. The Z-score formula is:
\[ Z = \frac{(X - \mu)}{\sigma} \] where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
In the exercise, Z-scores are used to convert dollar amounts (like \\(450) into a standard format. This helps in determining the position of a specific dollar amount relative to the entire distribution. For example, the Z-score of \\)450 is calculated as -1.384. This means \$450 is 1.384 standard deviations below the average spending.
Z-scores are crucial for understanding how unusual or typical a single observation is within the distribution, guiding decisions based on probability estimates.

**Probability** quantifies the likelihood of a specific event occurring. In the context of a normal distribution, it helps us assess how likely it is for certain spending values to occur. Given a specific Z-score, we can look up its corresponding probability in the Z-table or compute it using a calculator designed for normal distributions.
For instance, the probability that a randomly selected family will spend less than \\(450 is about 0.0832, or 8.32%. This indicates it's relatively uncommon for spending to be that low. Conversely, to find the probability between two values (like \\)500 and \\(750), we calculate the difference in probabilities for their corresponding Z-scores: 0.8686 for \\)750 and 0.1660 for \$500. This results in a 70.26% chance that a family will spend between these two amounts.
Understanding probability in the context of normal distribution allows for a refined analysis of data, predicting outcomes in practical scenarios with varying degrees of certainty.

A **random variable** is a variable whose possible values are outcomes from a random phenomenon. In the context of our exercise, the random variable represents the amount of money a family spends on back-to-college items.
This expenditure is considered random because while there's an average or expected amount, each family's spending is influenced by various unpredictable factors. The randomness means we cannot predict exactly what one family will spend, but we can talk about the distribution of spending across many families.
In this exercise, understanding that the expenditure is a normally distributed random variable allows us to apply statistical tools, like mean and standard deviation, to make probabilistic predictions. We then use these insights for informed decision-making about expected spending behaviors.