Question

Assume we have a box with 10 chips, each with a number written on it from 1-10.

We reach in, pull out a chip, look at the number and write it down, then put the chip back. We do
this 2 times. We describe this as N = 2 for the sample size. Then we take the mean of the two numbers
we recorded.

We repeat this 5000 times, each time drawing a random sample of 2, with replacement (i.e., N=2).
Each time we record the mean we observed. When we are done, we have 5000 means, each based on
two chips drawn at random. What do you expect the mean of these 5000 sample means to be? Why?

What do you expect the standard deviation of these 5000 means to be? Show your work.

Answer 1

Solution::

The probability of drawing a chip in the first draw i.e., N=1 is 1/10.

Since we put the chip back after it is drawn ,the probability of drawing a chip in the second draw i.e., N=2 will be 1/10.

Hence each chip has equal probability to get picked in both the draws. The minimum value on the chip can be 1 and the maximum value on the chip can be 10.

Therefore the mean of the two numbers cannot be less than 1 and also cannot be more than 10.

The mean can have values from 1 to 10 with each having equal probability.