Each sweat shop worker at a computer factory can put together 4.6 computers per hour on average with a standard deviation of 1 computers. 50 workers are randomly selected to work the next shift at the factory. Round all answers to 4 decimal places where possible and assume a normal distribution. a. What is the distribution of X? X - N( 4.6 1 b. What is the distribution of x? ¤ - N( 4.6 1414 o o c. What is the distribution of >x? ) x - N( 230 7.0711 v) o d. If one randomly selected worker is observed, find the probability that this worker will put together between 4.4 and 4.6 computers per hour. .0793 e. For the 50 workers, find the probability that their average number of computers put together per hour is between 4.4 and 4.6. .4214 f. Find the probability that a 50 person shift will put together between 225 and 235 computers per hour. .5205 g. For part e) and f), is the assumption of normal necessary? YesO No o h. A sticker that says "Great Dedication" will be given to the groups of 50 workers who have the top 10% productivity. What is the least total number of computers produced by a group that receives a sticker? minutes (round to the nearest computer)

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### Understanding Probability and Distribution in Manufacturing

In this example, we are looking into the productivity of workers at a computer factory. Each worker can assemble 4.6 computers per hour on average, with a standard deviation of 1 computer. In the analysis, we will consider a group of 50 workers chosen randomly. The answers assume a normal distribution and are rounded to four decimal places.

#### Questions and Answers:

**a. What is the distribution of \(X\)?**  
\(X \sim N(4.6, 1)\)

**b. What is the distribution of \(\bar{x}\)?**  
\(\bar{x} \sim N(4.6, 0.1414)\)

**c. What is the distribution of \(\sum x\)?**
\(\sum x \sim N(230, 7.0711)\)

**d. Probability for a single worker:**  
If one randomly selected worker is observed, find the probability that this worker will put together between 4.4 and 4.6 computers per hour.  
**Answer:** 0.0793

**e. Average productivity for 50 workers:**  
For the 50 workers, find the probability that their average number of computers assembled per hour is between 4.4 and 4.6.  
**Answer:** 0.4214

**f. Total productivity for 50 workers:**  
Find the probability that a 50-person shift will put together between 225 and 235 computers per hour.  
**Answer:** 0.5205

**g. Is the assumption of normal distribution necessary for parts (e) and (f)?**  
**Answer:** Yes

**h. Top 10% productivity reward:**  
A sticker that says "Great Dedication" will be awarded to groups of 50 workers who are in the top 10% of productivity. What is the minimum total number of computers produced by a group that receives the sticker?

Unfortunately, the necessary number for point h is not provided in the image and it must be determined using further statistical methods.

#### Explanation of Graphs and Diagrams:

No graphs or diagrams are provided in this example, but typically, this problem could be illustrated with standard normal distribution curves to visually represent probabilities and distributions regarding the workers' productivity.

This material provides a practical application of normal distribution in a real-world factory setting, helping students understand how to calculate and interpret probabilities
Transcribed Image Text:### Understanding Probability and Distribution in Manufacturing In this example, we are looking into the productivity of workers at a computer factory. Each worker can assemble 4.6 computers per hour on average, with a standard deviation of 1 computer. In the analysis, we will consider a group of 50 workers chosen randomly. The answers assume a normal distribution and are rounded to four decimal places. #### Questions and Answers: **a. What is the distribution of \(X\)?** \(X \sim N(4.6, 1)\) **b. What is the distribution of \(\bar{x}\)?** \(\bar{x} \sim N(4.6, 0.1414)\) **c. What is the distribution of \(\sum x\)?** \(\sum x \sim N(230, 7.0711)\) **d. Probability for a single worker:** If one randomly selected worker is observed, find the probability that this worker will put together between 4.4 and 4.6 computers per hour. **Answer:** 0.0793 **e. Average productivity for 50 workers:** For the 50 workers, find the probability that their average number of computers assembled per hour is between 4.4 and 4.6. **Answer:** 0.4214 **f. Total productivity for 50 workers:** Find the probability that a 50-person shift will put together between 225 and 235 computers per hour. **Answer:** 0.5205 **g. Is the assumption of normal distribution necessary for parts (e) and (f)?** **Answer:** Yes **h. Top 10% productivity reward:** A sticker that says "Great Dedication" will be awarded to groups of 50 workers who are in the top 10% of productivity. What is the minimum total number of computers produced by a group that receives the sticker? Unfortunately, the necessary number for point h is not provided in the image and it must be determined using further statistical methods. #### Explanation of Graphs and Diagrams: No graphs or diagrams are provided in this example, but typically, this problem could be illustrated with standard normal distribution curves to visually represent probabilities and distributions regarding the workers' productivity. This material provides a practical application of normal distribution in a real-world factory setting, helping students understand how to calculate and interpret probabilities
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