Suppose a research group is trialling a new production method for bike wheel, production can a volatile process,

and the group has decided to accept the fact that there will be a certain proportion of failures out of the total number of wheel produced.

However, before committing to the new process, the group would like to estimate the probability of failure by making a number of wheels. They have asked you how many wheels they should product to ensure they have a reasonably good idea of the probability of failure.

The engineers developing the new production method assure management that the probability of failure is somewhere between 1% and 20%, but they are unwilling to make any guarantees beyond this without testing the method first.

Suppose that we are considering three potential failure probabilities:

• θ=0.02
• θ=0.2

We also are considering three potential sizes for our test production run (i.e. the number of wheels group will make in test run):

• n=50
• n=200

For each combination of failure probability and number of cases printed, calculate the limiting distribution for the sample mean. You should calculate 4 limiting distributions in total