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Probability calculation helps us figure out the likelihood of an event occurring. In our shoe example, we want to know the probability of a shoe weighing more than 13 ounces. First, we calculate the z-score using the formula \( z = \frac{X - \mu}{\sigma} \). Here, \( X \) is our specific value (13 ounces), \( \mu \) is the mean (12 ounces), and \( \sigma \) is the standard deviation (0.5 ounces). When we substitute these values in, we get \( z = \frac{13 - 12}{0.5} = 2 \).

Next, we look up this z-score in a z-table, which tells us how much of the data lies below this score. However, we're interested in the probability of the shoe weighing more than 13 ounces, which is essentially the area to the right of \( z = 2 \) in the normal distribution curve. From the z-table, this area (or probability) is approximately 0.0228, or 2.28%. This means only about 2.28% of the shoes will weigh more than 13 ounces.

The z-score is a measure of how many standard deviations a data point is from the mean. It is crucial for understanding how unusual or typical a value is within a distribution. A z-score of 0 means the value is exactly at the mean, while a positive or negative z-score indicates how far and in which direction the value deviates from the mean.

To calculate a z-score, we use the formula \( z = \frac{X - \mu}{\sigma} \). This formula shows the difference between the value (\( X \)) and the mean (\( \mu \)), scaled by the standard deviation (\( \sigma \)). For instance, a z-score of 2 in our shoe exercise implies that the weight of 13 ounces is 2 standard deviations above the mean. In a normal distribution, z-scores help us find probabilities and percentiles, making them indispensable for statistical analysis.

Standard deviation is a vital statistic that measures how spread out values are in a dataset. A low standard deviation indicates that values are clustered closely around the mean, while a high standard deviation suggests they are more spread out. In the exercise, our shoe weights have a standard deviation of 0.5 ounces, meaning the weights don't vary much from the average of 12 ounces.

Finding the right standard deviation can be important for meeting statistical goals, like ensuring 99.9% of shoes weigh less than a certain value. In the exercise, by knowing what percentage of shoes need to be under 13 ounces, we work backwards using the z-score to adjust the standard deviation as needed. This process allows companies to hold product weights within desired limits, ensuring consistency and quality in manufacturing.

Percentiles indicate the relative standing of a value in a data set. They show the percentage of data points that fall below a particular value. The 99.9th percentile means 99.9% of data points fall below it, leaving only 0.1% above. Percentiles are particularly useful for setting performance thresholds or quality standards.

In part b of our shoe exercise, we needed to find a standard deviation that allows 99.9% of the shoes to weigh less than 13 ounces. We found the z-score corresponding to the 99.9th percentile in a z-table, which is approximately 3.09. Using this z-score, we can adjust either the standard deviation or mean to ensure that 99.9% of the shoes meet the weight criteria. This allows companies to confidently state that the vast majority of their products meet specified weight limits.