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In statistics, the normal distribution is a vital concept because it describes how values are distributed in many natural and human-made processes. When data follows a bell-shaped, symmetric pattern, we often say it is normally distributed. But, handling various normal distributions with different means and standard deviations can be tricky.

That's where the standard normal distribution helps. It's a special type of normal distribution with a mean of 0 and a standard deviation of 1. By converting any normal distribution to the standard normal distribution, calculations become more straightforward and more manageable. This is done using Z-scores, which essentially "standardizes" your data by transforming it into numbers that fit on a unified scale.

In the exercise, we're tasked with understanding a distribution of printer dot diameters. Converting these diameters into the standard normal distribution simplifies the steps needed to find probabilities.

Z-scores are a statistical tool used to understand how far a particular value is from the mean, measured in units of standard deviation. Imagine you have a value within your data set and you want to know how unusual or typical it is. Calculating the Z-score can tell you just that.

The formula for Z-score is relatively straightforward:

• It is calculated as: \( Z = \frac{x - \mu}{\sigma} \)

Here, \(x\) is your value of interest, \(\mu\) is the mean of the data, and \(\sigma\) is the standard deviation. The Z-score expresses how many standard deviations away \(x\) is from the mean. A negative Z-score indicates the value is below the mean, while a positive score indicates it is above.

In our exercise, we calculated Z-scores to convert diameter sizes to the standardized form required for looking up probabilities in the standard normal distribution table. For example, a diameter of 0.0026 inches had a Z-score of 1.5, implying it is 1.5 standard deviations above the mean.

When dealing with normal distributions, determining the probability of a certain value or range of values is a common task. Using the standard normal distribution and its associated tables (or software tools), we can easily find probabilities linked to Z-scores.

Here's how it works:

• First, compute the Z-scores for your values of interest.
• Next, use a Z-table, which gives you the probability of a random variable being less than your computed Z-score.
• If you need the probability of it being greater than the Z-score, you subtract the table value from 1 because the total probability in a distribution sums to 1.

For example, if you look up a Z-score of 1.5 and find that the table gives you 0.9332, the probability of having a dimension exceeding 0.0026 inches is 1 minus this value, giving 0.0668. These calculations let us understand how likely certain dimensions are to occur.

In our exercise, we're focusing on the diameter of dots produced by a printer, which naturally follows the properties of a normal distribution with a specified mean and standard deviation.

This distribution can tell us crucial things about the performance and consistency of the printer: whether it largely produces dots around the expected size and how much variation there is.

• A smaller standard deviation would mean most dots are close to the mean, implying greater consistency.
• Whereas a larger standard deviation means more variation in diameters.

Part (c) of the problem deals with finding a new standard deviation. This ensures a certain probability that diameters fall within a certain range. Such engineering problems make these statistical concepts a valuable tool in quality control and assurance, ensuring products meet desired specifications consistently.