You work for an engineering company that sells a machine that makes the coolest widgets. You have a potential client that wants to buy your machine, but he’s having doubts since the time it takes for the machine to make the widget varies every time. Fortunately you know that the time it takes to produce each widget is uniformly distributed between 4 and 10 seconds, independent of each other.

1. He is not convinced by your distribution argument and he wants to see "actual" data. You can prepare for him an experiment to try to assure him regarding the average time it takes to produce each widget. To do this you will make n widgets, measure how long it takes each one, and then calculate the sample mean. In other words, if you measure that it takes Ti seconds to make the ith widget, you will calculate:

Mn= (T1+T2 + ... + Tn)/ n

There is a lot at stake in this experiment so you want to ensure that the value you compute, Mn, is within 0.01 of the expected value of the time to make each widget. Use the Chebyshev inequality to calculate the number, n, of the widgets you should make to ensure the value is within the desired range with a probability of 95%.

2. You also want to convince him regarding the amount of time it will take to make 1000 widgets. In a separate experiment, you will make 1000 widgets and compute how long it took to make the 1000 widgets in total. You will tell you client that you can complete the 1000 widegets in 7050 seconds. Use the Central Limit Theorem to calculate the probability you will succeed.