Random Sampling and Estimation Part B: Selecting the Sample (30 minutes)

In This Part: Fair Sampling
You may have noticed that your estimates for the total penguin population vary quite a bit based on both the sample size and which sub-regions were sampled. The decision about how to select a sample, accordingly, is a critical one in statistics. It is important that each part of the population be treated fairly. If you are fair in the selection, then you should obtain a representative sample and thus a more fair estimation procedure.

In earlier sessions, you looked at notions of fairness and randomness and noticed that people have a difficult time being fair or random. So what methods can you use to accomplish fair sampling? See Note 4 below.

Problem B1
How might you select 10 sub-regions from the 100 total sub-regions so that you would be most likely to have a “representative” sample for estimating the size of the penguin population in the entire region? You can use the empty chart below to explore your ideas.

 To select 10 sub-regions from the 100 total sub-regions in a “fair” way requires that each of the 100 sub-regions has the same chance of being selected. You can accomplish this with random selection. How might you select 10 sub-regions in a random fashion?

Video Segment
In this video segment, groups of participants devise methods for collecting a random sample of penguins. Watch this segment after you have completed Problem B1 and compare your method with that of the onscreen participants. Do these methods ensure that the samples will be random?

In This Part: A Fair Sampling Method
There are many different ways to randomly select 10 sub-regions. Many of these methods involve initially numbering the 100 sub-regions. In this section, we will use the numbering system below, which numbers the sub-regions from 00 through 99:

Locating number positions is easier if we put digits on the outside borders as shown. Each number in the grid corresponds to a red and blue number combination; the red number is the first digit, and the blue number is the second digit:

Problem B2
Think of a way to pick 10 numbers between 00 and 99 at random. (You may prefer to select each digit individually, or to select the entire two-digit number at once.) Then use your method to generate the 10 random numbers.

 You may wish to use a random-number-generating device, such as a calculator, a 10-sided die, or computer software, to generate the random numbers.

One possible method for solving Problem B2 is to use two 10-sided dice, one red and one blue. The sub-region can then be determined by the two dice (in the order red, and then blue).

You might notice that the random selection process will sometimes produce duplicates. There is a greater than one-third chance that 10 numbers picked at random between 00 and 99 will produce at least one duplicate, and almost a 90% chance that 20 such numbers will produce at least one duplicate.

For instance, you might find that seven tosses of the dice produced these sub-region choices:
19 22 39 50 34 05 39

If we do not want duplicates, we can skip them until we get 10 distinct numbers, for example:
19 22 39 50 34 05 75 62 87 13

This is called sampling without replacement, since each time we choose a sub-region we remove it from the list of sub-regions we can choose on the next toss of the dice. In some experiments, it may be impractical or impossible to exclude duplicates from the random selection process. If duplicates are allowed, it is called sampling with replacement.

The 10 distinct numbers (19, 22, 39, 50, 34, 05, 75, 62, 87, 13) correspond to these 10 sub-regions:

Here is a look at the number of penguins in each of the 10 sub-regions we selected:

The estimate of the total number of penguins for the entire region based on this random sample of 10 sub-regions is as follows:

100 x [(5 + 6 + 6 + 7 + 5 + 2 + 1+ 5 + 5 + 3)/10] = 100 x (45/10) = 450

Problem B3
Use the random sample you found in Problem B2 to estimate the total number of penguins in the region. Find your 10 random sub-regions in the chart below:

Problem B4
Did you expect your estimate from Problem B3 to equal your estimate from Problem B2? Why or why not? What explains this variation? If the sample size were increased to 20 sub-regions, would you expect the variation in the estimates to increase or decrease? Why?

In This Part: Variation in Estimates
A computer can perform random sampling and estimation much more quickly than you can by hand. Here are three more random samples of 10 sub-regions generated by a computer.