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Normal distribution is a foundational concept in statistics. It's often referred to as a "bell curve" because of its symmetric shape, peaking at the mean with progressively decreasing tails. This distribution is characterized by its mean and standard deviation. The mean (\(\mu\)) represents the average of all data points, while the standard deviation (\(\sigma\)) measures the spread or variability of the data. Most data in a normal distribution falls within three standard deviations of the mean. This property is critical in various fields, including quality control, finance, and environmental studies. In this exercise, the printer dot diameter is normally distributed with a mean of 0.002 inches and a standard deviation of 0.0004 inches.

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed in terms of a number of standard deviations from the mean. Calculating the Z-score helps us understand how far or how close a particular data point is to the average.

To calculate a Z-score, use the formula:

• \[Z = \frac{X - \mu}{\sigma}\]

Where:

• \(X\) is the value to be standardized,
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

For example, to calculate the Z-score for a dot diameter of 0.0026 inches:

• \[Z = \frac{0.0026 - 0.002}{0.0004} = 1.5\]

The Z-score of 1.5 signifies that 0.0026 inches is 1.5 standard deviations above the mean. Calculating Z-scores enables us to find probabilities using the standard normal distribution tables or a calculator.

Standard deviation is a measure of how spread out numbers are around the mean. A smaller standard deviation indicates that the data points are close to the mean, whereas a larger standard deviation signifies more dispersed data. This measure is crucial when comparing variability between datasets that might have different averages.

To find a new standard deviation that changes the probability as needed, adjustments are made using the Z-score relationships and desired probabilities. In the exercise, we sought a situation where the probability of the diameter falling between 0.0014 and 0.0026 inches is 0.995.

To accomplish this, the approach is to use the existing normal distribution mechanics, determining \(\sigma'\), the adjusted standard deviation, as follows:

• Find the necessary Z-score for the target probability, e.g., for a right tail probability of 0.0025 leads to Z = 2.807.

Then use the desired range:

• \[\sigma' = \frac{0.0026 - 0.0014}{2 \times Z}\]
• \[\sigma' \approx 0.000214\]

This adjustment ensures the data conforms to the confidence levels required by the setting, which is a key use of standard deviation in statistical assessments.