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The mean is a central concept in statistics and refers to the average value of a dataset. It is calculated by summing all the values in the dataset and then dividing by the number of values. In the context of the normal distribution of wiring packages, we start with an hourly mean of 16.4 packages per hour. For an entire 8-hour workday, we calculate the mean by multiplying the hourly mean by the number of hours worked: \( \mu = 16.4 \times 8 = 131.2 \).
This value, 131.2 packages, represents the average number of packages a worker is expected to assemble in an 8-hour day under normal conditions. Understanding the mean helps us identify the expected average workload and can guide scheduling and resource allocation to meet production targets. This average is crucial for predicting outcomes and making informed management decisions.

Standard deviation is a measure of how spread out numbers are in a dataset. It tells us how much variation exists from the mean. A low standard deviation means that the values are close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range. In this exercise, the standard deviation per hour is given as 1.3 packages. For an entire 8-hour day, the standard deviation changes as more hours offer more variability.
The formula to calculate the standard deviation over multiple periods for independent tasks is \( \sigma' = \sqrt{n} \times \sigma \). For 8 hours, this becomes \( \sigma' = \sqrt{8} \times 1.3 \).
Simply put, standard deviation helps assess the reliability of the mean by illustrating the extent of deviation in package production, influencing quality control and efficiency metrics.

A probability distribution describes how probabilities are distributed over the values of a random variable. It provides a complete description of the likelihoods associated with every possible outcome of a random experiment. In the case of the assembled wiring packages, the normal distribution is used with a specified mean and standard deviation.
Normal distributions are symmetrically mound-shaped and can be encountered frequently in nature. When we modify the conditions—changing from hourly production to an 8-hour day—the overall shape remains "normal" due to the properties of normal distribution, meaning probabilities remain predictable and well-calibrated. Understanding the concept of probability distribution lets us predict and quantify the variance in outcomes like production rates and quality assurance.

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values. It is expressed as the number of standard deviations a data point is from the mean. For an assembled package calculation, the Z-score helps determine how unusual a production rate (e.g., 135 packages) is compared to the expected rate.
Calculating the Z-score involves using the formula \( Z = \frac{x - \mu'}{\sigma'} \), where \( x \) is the number of interest (in this case, 135 packages), \( \mu' \) is the mean for 8-hour day, and \( \sigma' \) is the standard deviation for 8-hour day.
This Z-score is then used to find probabilities in a Z-table, which reveals how likely it is for a worker to produce at least 135 packages in a single day. Understanding Z-scores is fundamental when assessing production targets and ensuring workforce capabilities are strategically aligned with expected outputs.