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The Z-score is a statistical measurement that describes a value's relation to the mean (average) of a group of values. When you use the normal distribution, the Z-score tells you how many standard deviations an element is from the mean of the dataset. This is a useful tool for determining the position of a data point within a normal distribution.

Let's break down the formula:

• Given by \( Z = \frac{X - \mu}{\sigma} \)
• \( X \) is the value for which you're finding the Z-score.
• \( \mu \) is the mean of the data.
• \( \sigma \) is the standard deviation.

By calculating the Z-score, you can easily compare different data points within the same distribution, even if they're not on the same numerical scale. In our exercise, calculating the Z-score for the budgeted $646, we use \( \mu = 615 \) and \( \sigma = 42 \). The Z-score calculation tells us where the budgeted amount lies in relation to the restaurant's typical weekly spending.

When working with normal distributions, a primary focus is to calculate probabilities using the Z-score. This is where the Z-score conversion of our data is crucial. By transforming our values into Z-scores, we can reference standard Z-tables to find probabilities related to our value of interest.

For instance, in our task, once the Z-score for \(646 is calculated as approximately \(0.74\), we then use a standard normal distribution table:

• Find \( P(Z \leq 0.74) \), the probability of a value being less than or equal to this Z-score, typically around \(0.7704\).
• To find the probability of exceeding \)646, calculate \( P(X > 646) = 1 - P(Z \leq 0.74) \).
• This gives us a probability of approximately \(0.2296\).

These steps demonstrate how understanding and calculating probabilities through Z-scores allows for accurate predictions, quite beneficial in budgeting contexts as explored in our exercise.

Budgeting involves predicting and planning for expenses in order to ensure financial stability. In the context of our exercise, a powerful application of Z-scores and probability calculations is creating a solid budgeting strategy that reduces risk.

To decide on a budget that meets a particular risk tolerance level (in this case, a 0.10 or 10% chance of being exceeded), you can reverse-engineer the problem:

• We first determine the Z-score for the desired probability: \( P(Z \leq z) = 0.90 \) which corresponds to \( z \approx 1.28 \).
• The formula \( Z = \frac{B - \mu}{\sigma} \) is then used to solve for \( B \), the budget amount.
• By rearranging the formula and substituting \( \mu = 615 \), \( \sigma = 42 \), and \( z = 1.28 \), we calculate \( B \approx 669 \).

This calculated budget helps ensure that the probability of needing to exceed the set budget is only 10%, promoting a balanced and informed approach to financial planning for the restaurant's weekly operations.