Question

The number of wiring packages that can be assembled by a company's employees has a normal distribution, with a mean equal to 16.4 per hour and a standard deviation of 1.3 per hour. a. What are the mean and standard deviation of the number \(x\) of packages produced per worker in an 8-hour day? b. Do you expect the probability distribution for \(x\) to be mound-shaped and approximately normal? Explain. c. What is the probability that a worker will produce at least 135 packages per 8 -hour day?

Short answer

Answer: Part a: In an 8-hour day, the mean number of packages produced per worker is calculated as: Mean for 8-hour day: μ = 16.4 × 8 The standard deviation for an 8-hour day is calculated as: Standard deviation for 8-hour day: σ' = √n × σ Part b: The probability distribution for the number of packages produced per worker in an 8-hour day is expected to be mound-shaped and approximately normal. Part c: To calculate the probability of a worker producing at least 135 packages in an 8-hour day, first find the Z-score using the formula Z = (x - μ')/σ'. Then, refer to the Z-table to determine the probability of a worker producing less than 135 packages in an 8-hour day. Subtract this probability from 1 to find the probability of a worker producing at least 135 packages: P(X ≥ 135) = 1 - P(Z ≤ Z-score)

Step-by-step solution

(Part a: Calculate mean and standard deviation per 8-hour day)

First, to find the mean and standard deviation per worker in an 8-hour day, we need to adjust the given values for an 8-hour day. Since the workers produce packages at an average rate of 16.4 packages per hour, the mean for an 8-hour day can be found by multiplying the hourly mean by the number of hours worked: Mean for 8-hour day: \(\mu = 16.4 \times 8\) Standard deviation is also affected by the number of hours worked. The formula for the standard deviation of a sum of independent variables is: Standard deviation for 8-hour day: \(\sigma' = \sqrt{n} \times \sigma\) Here, \(n\) is the number of hours worked and \(\sigma\) is the hourly standard deviation.

(Part b: Distribution shape explanation)

Since the number of wiring packages assembled has a normal distribution, the probability distribution for \(x\) (number of packages produced per worker in an 8-hour day) is expected to be mound-shaped and approximately normal. This is because normal distributions typically maintain their shape when subjected to arithmetic operations.

(Part c: Probability calculation)

To calculate the probability that a worker will produce at least 135 packages per 8-hour day, we'll need to use the Z-score formula. The Z-score, in this case, would represent how many standard deviations above the mean the desired value (135 packages) is. The Z-score formula is as follows: \(Z = \frac{x - \mu'}{\sigma'}\) We have already calculated \(\mu'\) and \(\sigma'\) in part a. With \(x = 135\), we can now find the Z-score. Next, we'll refer to the Z-table to determine the probability of a worker producing less than 135 packages in an 8-hour day. To find the probability of a worker producing at least 135 packages, we'll subtract the Z-table probability from 1. P(X ≥ 135) = 1 - P(Z ≤ Z-score)

Key concepts

Mean

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

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.

Probability Distribution

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.

Z-score

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.