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The Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. If you're wondering what it exactly means, think of it as a way to see how far a number is from the average of a data set, in terms of standard deviations. For our semiconductor line width example, calculating the Z-score allows us to transform the 0.62 micrometer line width into a format suitable for comparison using a standard normal distribution table.
To calculate the Z-score, use the formula:

• \[Z = \frac{X - \mu}{\sigma} \]

Where:

• \(X\) is the data point of interest,
• \(\mu\) is the mean, and
• \(\sigma\) is the standard deviation.

By transforming the data point into a Z-score, we can easily assess if the measurement is above or below the mean and by how much.

Finding the probability is about determining how likely a certain event is to happen. In the context of our exercise, we're looking for the probability of certain line widths occurring in the semiconductor manufacturing process. By converting data values into Z-scores as previously described, we can use the standard normal distribution (also known as the Z-distribution) to find these probabilities.
Let's remember the steps: - **Calculate Z-scores**: Once you have the Z-scores, you can easily refer to Z-tables which detail the cumulative probability of a standard normal distribution. - **Look Up in Table**: For part (a), the probability for a line greater than 0.62 micrometers is found by calculating the area to the right of the corresponding Z-score (1 - cumulative probability). - **Calculate Range Probabilities**: For part (b), calculate the probability for two Z-scores and subtract the smaller from larger to find the probability between the values. The use of Z-scores and probability tables allows us to translate real-world values into probabilities, which is crucial for quality control in manufacturing.

Percentiles indicate the relative standing of a value within a data set, showing the percentage below which a given value falls. In semiconductor manufacturing, knowing the percentile of line widths can help in maintaining the quality and functionality of semiconductors. For example, part (c) of our exercise asks for the 90th percentile—this is essentially asking what line width is greater than 90% of all the measurements. To calculate it: 1. **Determine the Z-score**: Using percentile values, consult a Z-table to find the Z-score representing the desired percentile (in this case, Z = 1.28 for the 90th percentile). 2. **Calculate the Line Width**: Apply the Z-score formula again to solve for the line width corresponding to this Z-score. The benefit of understanding percentiles is that it allows manufacturers to set specifications that ensure most products will fall within an acceptable range, thus maintaining high standards in production processes.

Semiconductor manufacturing is a complex process that requires precision, with line widths of circuits being a vital attribute. Keeping these widths consistent ensures the functionality and reliability of semiconductor devices, like memory chips and processors. Variations in line width can lead to different operational characteristics which might be undesirable. Statistically controlling these line widths is achieved through understanding and applying normal distribution principles. This helps in: - **Identifying deviations**: By knowing how much deviation from the mean is acceptable. - **Quality assurance**: Ensuring that 90% or more of products meet minimum standards, as indicated by a lower percentile threshold. - **Process optimization**: Using these insights to make informed decisions in designing and improving the manufacturing process. Thus, the application of normal distribution, Z-scores, and probability helps semiconductor manufacturers maintain high-quality production and stay competitive in the high-tech market.