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To understand quality control in the context of normal distribution, it's essential to get familiar with Z-score calculation. The Z-score helps us determine how many standard deviations a data point is from the mean.

The formula for calculating the Z-score is: \(Z = \frac{X - \mu}{\sigma}\) where:

• X is the data point (in this case, the shell thickness of 0.025 mm)
• \(\mu\) is the mean (0.03 mm)
• \(\sigma\) is the standard deviation (0.0015 mm)

In our example, substituting the values in gives: \(Z = \frac{0.025 - 0.03}{0.0015} = \frac{-0.005}{0.0015} = -3.33\). This result tells us that the thickness of 0.025 mm is 3.33 standard deviations below the mean.

Understanding Z-scores is essential because they allow us to use standard normal distribution tables to find probabilities.

Probability theory provides the frameworks to understand and evaluate events' likelihood. When a data set follows a normal distribution, it means we can use Z-scores and related tables to find probabilities associated with specific ranges of data.

For instance, in our quality control scenario with the toy manufacturer, we calculated a Z-score of -3.33 for a ball thickness of 0.025 mm. With this Z-score, we can determine the probability of this event using a standard normal distribution table.

By looking up -3.33 in the Z-table, we find a cumulative probability of approximately 0.0004. This small probability (0.04%) indicates it is quite unlikely to find a ball with a shell thickness less than 0.025 mm.

This insight allows manufacturers to assess quality control measures effectively and ensures products stay within specified tolerances.

Standard deviation is a measure of the dispersion or spread of a set of values in a data set. It describes how much individual data points differ from the mean. The formula is: \[\sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}}\] where:

• X_i is each data point
• \(\mu\) is the mean
• N is the number of data points

In the context of quality control for the toy manufacturer, a standard deviation of 0.0015 mm indicates minor variability in the shell thickness of hollow rubber balls. This small standard deviation shows the precision in production, providing confidence that most balls produced will have a thickness close to the mean of 0.03 mm.

Accurate standard deviation helps in setting control limits and ensures the manufacturing process adheres to our quality standards.