Continuous Probability Distribution \[ \begin{array}{l}\text { Discussion Topic } \\ \text { Question: Imagine you're responsible for quality control in a factory that produces metal rods with lengths uniformly } \\ \text { distributed between } 95 \mathrm{~cm} \text { and } 105 \mathrm{~cm} \text {. } \\ \text { 1. Probability calculation: What is the probability that a randomly selected rod measures between } 98 \mathrm{~cm} \text { and } 102 \mathrm{~cm} \text { ? } \\ \text { 2. Quality specification: If rods shorter than } 96 \mathrm{~cm} \text { or longer than } 104 \mathrm{~cm} \text { are considered defective, what proportion of } \\ \text { the production is expected to be defective? } \\ \text { 3. Process Adjustment: To reduce the defect rate, management is considering adjusting the production process to } \\ \text { produce rods uniformly distributed between } 97 \mathrm{~cm} \text { and } 103 \mathrm{~cm} \text {. How would this change impact the proportion of } \\ \text { defective rods? }\end{array} \]

Let’s work through each part step by step. 1. Probability that a randomly selected rod measures between 98 cm and 102 cm:  • The rods are uniformly distributed between 95 cm and 105 cm. The total length of the interval is 105 – 95 = 10 cm.  • The interval of interest (98 cm to 102 cm) has a length of 102 – 98 = 4 cm.  • For a uniform distribution, the probability is proportional to the interval length:   Probability = (Length of the interval of interest) / (Total length) = 4/10 = 0.4 or 40%. 2. Proportion of production expected to be defective when rods shorter than 96 cm or longer than 104 cm are deemed defective:  • Defective if rod < 96 cm or rod > 104 cm.  • For rods less than 96 cm:   Interval from 95 cm to 96 cm, length = 96 – 95 = 1 cm.  • For rods greater than 104 cm:   Interval from 104 cm to 105 cm, length = 105 – 104 = 1 cm.  • Total defective length = 1 cm + 1 cm = 2 cm.  • Defective proportion = (Total defective length) / (Total interval length) = 2/10 = 0.2 or 20%. 3. Impact of adjusting the production process so that rods are uniformly distributed between 97 cm and 103 cm:  • New distribution interval is now from 97 cm to 103 cm, so the total length is 103 – 97 = 6 cm.  • Quality specification remains that rods are acceptable if they are between 96 cm and 104 cm (rods shorter than 96 or longer than 104 are defective).  • Notice that the new range [97, 103] lies completely within the acceptable range [96, 104].  • Therefore, under the new process, every rod produced falls within specification, and the defect rate becomes 0%. In summary: 1. The probability that a rod is between 98 cm and 102 cm is 40%. 2. Under the original process, 20% of the rods are expected to be defective. 3. With the adjusted process, none (0%) of the rods are defective relative to the 96–104 cm quality standard.