Problem $1$ Background : Line balancing is a crucial concept in fean manufacturing that focuses on distributing and timing work evenly across assembly flow lines to eliminate bottlenecks and excess capacity Balanced production lines mean increased productivity $,$ reduced lead times $,$ and considerable cost savings Knowledge of resource requirements including time $,$ is important in designing processes that are balanced and free of bottlenecks $.$ Problem : A manufacturing company is evaluating its staffing levels $.$ In particular $,$ the number of employees needed for the various assembly stages is under consideration $.$ Part of the evaluation process involves determining how much time must be allowed to complete the specific task at each stage $.$ Company management want to know the maximum amount of time that should be allowed to complete each task $95\%$ of the time $.$ Data has been provided for the tasks at two stages $,$ A and B$.$ Each set of data should be regarded as a random sample of $250$ recent times $($ in minutes $)$ that were required to complete the task $.$ This data can be found in the Excel worksheet titled "Case $1$ Part I Times $.$ Requirements : $1.(4$ points $)$ Summarize each set of data $($ Times Times B $)$ graphically and numerically Your numerical summary should include the sample size $,$ the $5-$number summary as well $,$ the average $,$ and standard deviation Use StatCrunch $,$ Excel $,$ or other suitable statistical computing software to ease your computational burden $2.(4$ points $)$ Based on the summary information obtained in requirement $1,$ choose a model to describe each set of data $($Times A Times B $)$ Examples of three models for continuous data are given on the next page $.$ The times to complete each of task A and task B can be modeled well by one of these distributions The examples were not generated from case data $.$ Each of your models will have one of these shapes but different parameters $.3.(3$ points $)$ Use the models to obtain the information that company management desires $($ percentile of time required for each task $.$ You will not use the PLAN $-$DO $-$ REPORT approach to writing up this problem $.$ Simply respond to each of the requirements in the order stated