Question: A company is investigating its average manufacturing costs per item produced. It has chosen a random sample of days and calculated the daily average manufacturing cost per item and has recorded the number of items produced. These are given in the table below:

The company's accountant believes the relationship between average cost and items produced is quadratic and further believes when no items are produced then the average cost is zero.

a) Write down an equation for the relationship between average cost and items produced as assumed by the accountant.

b) Derive, from first principles, least squares estimators for the parameters in this model

c) Calculate estimates of these parameters from the data above.

d) Why is it important that the data was from a random sample?

e) Without carrying out any formal tests, discuss whether the accountant's assumed model is reasonable.

f) An alternative model proposed is

\[y=\alpha +\beta {{x}^{\gamma }}\]

where y is average cost and x is items produced. Briefly explain the problems inherent in trying to find least squares estimates of the parameters in this model.