Sampling and Making Predictions

Objective: When done with this lesson, you will have demonstrated how to

explain a method for solving a real-world problem involving direct proportion
apply a method for solving a real-world problem involving direct proportion

Approximate completion time: 2 hours

Have you ever wondered how biologists estimate the number of fish in a lake or bears in the wild? Certainly they don't count every one. That would be nearly impossible and dangerous too! A smart biologist would use proportional reasoning.

Let's try an example of our own.

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Suppose you wanted to know approximately how many bees were in a hive. Obviously you don't want to count every single bee.

Solution:
Take a small sample of the bee hive. Let's presume that sample had 30 bees in it. On each of the bees in the sample, make a small identifiable mark with a white marker. (Radio collars or ear tags wouldn't so work well for small bees.) These "tagged" bees will then be released back into the larger population of the hive.

After some time under which you believe the tagged bees from the sample have mixed back in with the population of the hive, take another sample. Let's presume this second sample had 80 bees in it. In this second sample there will likely be some of the tagged bees from the first sample. Suppose there were 10.

So in effect you would have the following proportion:

Which for our example would be , where x is the total bees in the population of our hive.

Think about it. If 1/8 of the bees in the second sample were tagged. Then if that sample were a good representation of the whole, wouldn't 1/8 of the total population of the hive be tagged?

Solving , I see that 30·8 = 240. So x = 240. (That is, I used the numerator to denominator comparison for equal ratios.)

Therefore, from our sample we predict that there are 240 bees in the hive.

Note: While many people dislike bees (not wasps, which many people lump into the same category), their role in our agricultural economy cannot be underestimated. Without the pollination that is best done by bees, the yields of many crops like apples, pears, cherries, blueberries and cranberries would be decimated. For more information visit sites like cyberbee.net. Bee sure to visit the BeeResearch page and the bee researchers.
(And yes, bees really do getted tagged .)

Investigation 1:

In this investigation you will practice sampling and making predictions. You will predict the number of beans in a population of beans. (Of course you may substitute something else for beans if you want.)

Materials you will need:

A population of beans. This would be a large amount you don't already know.

A container for your beans. A brown paper bag or an empty butter dish would work well.

A marker to tag the beans.

Some scratch paper to record your results.

Steps:

Place a large amount of beans into your container. Close the container and let the beans "mingle". This container of beans is your "population of beans". Be sure to put any unused beans away.

Take a small sample of beans from the population of beans. You should have at least 10 and less than 50. Count and record how many are in this sample.

Using a marker, tag each of the beans in your sample. You can use a small "X" or some other distinguishable mark.

"Let the beans go." Put the tagged beans back into the container with the rest of the population.

Let the beans mingle and go about their business. (Of course, you may need to shake the container to help out.)

Take a second sample from the population. Record how many are in this sample and how many in this sample are tagged.

Complete this proportion and solve for x, the total beans in your population.

Place the beans from your second sample back into the population. Let the beans mingle again. Take another sample and repeat the proportion from #7 with this time calling it "sample #3".

Repeat the sampling procedure one more time for a total of three predictions on the total population of your beans.

Using the results of your three predictions, make a guess at how many beans are actually in the population. If you'd like to know how close you are, then count all the beans in the population.

Investigation 2: (Optional)

Have you ever wondered how many chocolate chips are in a bag of chocolate chips?

Materials you will need:

A bag of chocolate chips. (If you dislike chocolate chips, then find your own substitution.)

Supplies to make chocolate chip cookies. (Here's a page of recipes if you need.)

A different chip to use as a "tagged" chocolate chip. (Butterscotch chips get my vote, but M&M's would suffice.)

A "sampling" crew.

Some scratch paper to record your results.

Steps:

Take a small sample, 10-30 chips, from your bag of chocolate chips. Record how many are in this sample.

Replace those chips you removed from the bag in step #1 with some "tagged" chips that can be easily identified. Be sure to use something of similar size so as to not distort the results. A different flavored chip, like butterscotch, would work well. Be careful to place these tagged chips back into the bag so that the number of chips in the bag is back to its original amount. (Of course those chocolate chips that were removed will need to be "disposed".)

Make some chocolate chips cookies! Be sure to mix your tagged chips into the population sufficiently. It is fine if you do not want or need to use the entire bag of chocolate chips as you will be, in effect, taking a sample from the bag.

BE CAREFUL TO COUNT EVERY CHIP YOU EAT.
"Sample" some cookies after they are baked. Record how many chips are in your sample and how many of those are "tagged". Whenever you "sample", be sure to record the number of chips in the sample and how many of those are tagged.

After "sufficient" samplings, estimate the total number of chocolate chips that were in your original population by using proportions.

Note that predicting is only that. The accuracy of predictions depends on many factors. To know how accurate a prediction should be, one would have to venture deeper into statistics than this course allows. However, you can start you thinking in that direction by asking some basic questions like these:

How does the sample size effect the accuracy of the prediction?
How representative was the sample of the entire population?
What factors could have influenced what was collected in the samples?
Were there elements of bias in the techniques used that influenced the results? (Maybe you saw the butterscotch chips and thus chose that cookie....)

Here are some practice problems to help you with "Sampling and Making Predictions".