Question

A week’s production of galvanized nails was of uneven quality, due to a midweek adjustment of the coating process. In fact, the thickness of the coating was of two sizes:

0.12 mm for 70% of the nails

0.10 mm for 30% of the nails

a. Calculate the mean thickness µ .

b. To estimate µ , a sloppy sample of 100 nails was drawn. By “sloppy,” we mean that there wasn’t proper care taken so that the sample is representative of the population. In particular, the nails were not stirred before sampling. Since the nails with the thick coating tended to settle at the bottom of the pile (they are heavier), in the sample there was only a 50-50 chance that each nail drawn would have a thick/thin coating, rather than the 70-30 chance of the population. Compute the expected value of the sample mean computed from this sloppy sample.

c. As an estimator of µ , what is the bias of this sloppy sample mean?

Answer 1

Answer a :-

In the first case there is a sampling unit of 100 units

Therefore n =100

Now this means 70% units are of 0.12mm thickness while the remaining units are of 0.10mm thickness

Mean =X /n

Where X is the sum of all variables

=(0.12*70)+(0.10*30)

=8.4+4=11.4

Mean =11.4/100

=0.114

Answer b :-

Due to settling of heavy coating nails in the bottom the chances of getting the nails equates to 50% each .

This means

X =(0.12*50)+(0.10*50)=6+5=11

Mean =11/100=0.11

Answer c :-

Bias of sloppy sample means that a sample is collected in such a way that some units of the intended sample are less likely to be included than other units .

In the given example the bias is that the heavy coating nails which are more in number are settled below the lighter ones and thus creating a sampling bias of equal chance despite of the fact that light weighted nails are less in number as compared to heavyweight nails .