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A Z-score, also known as a standard score, is a statistical measurement that describes a data point's relation to the mean of a group of values. It represents how many standard deviations an element is from the mean. This makes it a very handy tool for comparing different distributions.
The formula to calculate the Z-score is:

• \[ Z = \frac{(X - \mu)}{\sigma} \]
• where:
  • \(X\) is the value in the dataset,
  • \(\mu\) is the mean value of the dataset, and
  • \(\sigma\) is the standard deviation of the dataset.

Knowing the Z-score helps to understand which percentile a certain value falls into within the normal distribution. A Z-score of 1.645, for example, suggests that the sales data point falls into the top 95% of data points in a typical normal distribution.

The mean and standard deviation are key components in understanding normal distributions. The mean is simply the average of all data points in a dataset and gives a central value to the dataset that represents its balance point.
To calculate the mean, you sum all the numbers and divide by the number of data points:

• \(\mu = \frac{\Sigma X}{N} \)
• where:
  • \(\mu\) is the mean,
  • \(\Sigma X\) is the sum of all data points, and
  • \(N\) is the number of data points.

Standard deviation, on the other hand, measures how spread out the numbers in a dataset are. A small standard deviation indicates that the values in a dataset tend to be close to the mean, while a large standard deviation indicates more spread out. Understanding both the mean and standard deviation allows you to interpret the shape and spread of data within a distribution effectively.

Calculation of inventory levels using the normal distribution helps businesses like manufacturers to ensure they have enough supply to meet demand, while minimizing the chance of running out of stock. Based on the manufacturer's requirement of only a 5% chance of stock shortage, one must find the right inventory level.

• To start, identify the area (in percentage) corresponding to the acceptable probability of running out, which in this case is 5% lacking stock, translating to a 95% confidence level.
• Next, obtain the Z-score from statistical tables for this confidence level, which is 1.645 for 95% of the standard normal distribution.
• Then, apply the formula for translating the Z-score into an inventory quantity: \[ X = \mu + Z \times \sigma \] where \(X\) is the desired inventory level.

Substitute the known values: - \(\mu = 1200\) (mean monthly sales), - \(Z = 1.645\), and - \(\sigma = 225\) (standard deviation of sales).
Thus, the calculation becomes: \[ X = 1200 + 1.645 \times 225 \] \[ X = 1200 + 370.125 \] \[ X = 1570.125 \] Finally, round to the nearest whole number to get a practical inventory level of 1570 mufflers, ensuring a stockout probability of just 5%.