If a random sample of size 16 is taken with replacement from the population 1, 1, 1, 2, 4, 5, 5 and 6, what is the mean, variance and standard deviation of the sampling distribution?

Question

Find the following:

 

* MEAN OF SAMPLING DISTRIBUTION

* STANDARD DEVIATION OF THE SAMPLING DISTRIBUTION

* VARIANCE OF THE SAMPLING DISTRIBUTION

Answer 1

Imagine an 8-segment spinner with segments labelled according to the data in the population dataset. Spinning this is equivalent to replacement after selection.

So the population (P) is infinite in size because of replacement. Now apply the population distribution to the population 100 (which we call R, the representative population) from which the sample of 16 will be taken.

⅜ of P are 1s, so in R there will be ⅜×100=37.5 1s, and so on for the other numbers: 12.5 2s, 4s and 6s, 25 5s. (Never mind that these are not all whole numbers. 37.5+12.5+12.5+25+12.5=100.)

The mean of P and R is 3.125. Their common variance (V) and standard deviations (SD) are about 3.859 and 1.984, the estimated V and SD for the sample are 3.859/16=0.241 and √V=0.491, or about 0.5. The mean of the sample is 3.125, the same as for P and R.