# Activity: Estimation

This activity is about estimation and doing a Survey.

## How many red marbles?

For this experiment you will need a bag of marbles of various colors - red, blue, green, yellow etc. The total number of marbles is not really important, but you will find it easier to work with a number like 50. What is important is that you do not know how many of each color ther are. So you will have to ask a friend or relative to choose the marbles for you.

You are going to estimate how many red marbles there are in the bag.

### So why don't you just count them?

Of course you could, but this experiment will teach you how to make an estimate. Scientists and statisticians use this method all the time to make estimates in real world situations where they can't just simply count. For example, there are four different human blood types, called A, B, AB and O. It would obviously be impractical to test everybody in the world; so, to find out how many people in the world have each blood type, scientists take a sample of (say) 1,000 people and test their blood type. From the sample they can estimate how many of each type there are in the world.

## Now you're ready to begin the experiment. It's very simple.

• Shake the marbles in the bag.
• Take one marble without looking and record its color.
• Return the marble to the bag.
• Repeat this process 100 times.

## Why do you return the marble to the bag?

The reason for this is so that the conditions for choosing the second marble are exactly the same as the conditions for choosing the first marble. If the first marble you chose was red and you didn't replace it, then for your second choice there would be one less red marble in the bag and one less altogether. That would make the chances of choosing another red marble different from the first time.

## How do you record the color?

You can use a tally/frequency table like this:

 Color Tally Frequency Red Blue Green Yellow Purple Total = 100

I have included five colors in my table, but yours could be different.

Once you've completed choosing 100 times, you can work out the relative frequency of red by dividing the number of red marbles by 100.

So, for example, if you recorded 22 red marbles in your table, then the relative frequency of red would be 22/100 = 0.22

## How does this help you estimate the number of red marbles in the bag?

That's easy. If there are 50 marbles in the bag and red occurred 22 times out of 100, then it should occur 11 times out of 50.

In other words, the fractions 22/100 and 11/50 are equivalent fractions.
An easier way is just to multiply 0.22 by 50: 0.22 × 50 = 11

When you've finished the calculations, you can check how many red marbles there really were in the bag.

• How good was your estimate?
• How can you get a better estimate?

To get a better estimate, don't count straight away.
Instead, repeat the experiment several times and calculate the mean number of red marbles.

Then you can compare your mean number with the actual number in the bag.
You should get a much better estimate, and may even get it exactly right.

As a variation on this experiment, you could use Smarties or M and M's. The only problem with that would be that you would probably be tempted to eat some and ruin the results of your experiment!

## How many A's?

Another variation on the above experiment can be done using the letters from the game of Scrabble. In the game of Scrabble there are 100 tiles. 98 of the tiles are inscribed with letters of the alphabet, the other two are blank. You don't have to remove the two blank tiles, but you might prefer to do so. I'll explain why later.

(If you don't have the game of Scrabble, then you could make your own tiles.
You will need 98 square pieces of cardboard, all the same size - 2 cm by 2 cm will do.

### Write letters on the tiles as follows:

• 9 A's
• 2 B's
• 2 C's,
• 4 D's,
• 12 E's,
• 2 F's,
• 3 G's,
• 2 H's,
• 9 I's,
• 1 J,
• 1 K,
• 4 L's,
• 2 M's,
• 6 N's,
• 8 O's,
• 2 P's,
• 1 Q,
• 6 R's,
• 4 S's,
• 6 T's,
• 4 U's,
• 2 V's,
• 2 W's,
• 1 X,
• 2 Y's
• and 1 Z

If you do this, then you will have to pretend that you don't know how many A's there are.

Now do the experiment in the same way that you did the marbles experiment.
Record your results in a table as before:

 Letter Tally Frequency A B C D E etc Total = 100

The only difference is that you will need a much longer table going all the way to Z.

When you've finished, calculate the relative frequency of the letter A and estimate the number of A's in the bag by multiplying the relative frequency by 98.

Do this several times and claculate the mean.

## So, why did we leave the two blank tiles out?

You can use your result to estimate the relative frequency of the occurence of the letter A in the English language. The manufacturers of the game of Scrabble have included 9 tiles with the letter A because (to the nearest whole number) the letter A occurs 9 times out of every 98 letters in the English language.

## How many people have blood type O?

Now you've learned how to make an estimate from a sample, you are ready to do a real-life experiment.
Before you begin you should read the pages How to Do a Survey and Survey Questions.

## How many people in the world have blood type O?

There are four different human blood types, called A, B, AB and O. It is obviously impractical to test everybody in the world; so, to find out how many people in the world have each blood type,you are going to take a sample of 100 people and find out their blood type. From the sample you will be able to estimate how many people in the world have blood type O.

There's only one problem with this experiment - many people don't even know their own blood type! And, since you can't test them for blood type, you will have to just count the ones who do know. Your sample, therefore, will be more than 100, so just take the first 100 who do know their blood type.

Note: Depending on where you live in the world, you could use a different physical characteristic such as hair color or eye color. But, if you live in some countries, you may not have a good variation of different colors. That's why I chose blood type.

## Which sample?

If you are still at school, then you could use your school population do do your survey. Ask permission first, then go around different classrooms. But you will probably find that  teenagers are even less likely to know their blood type than adults.

So...

To get a better result you will probably have to do a survey in your local shopping center or mall. When you stop people, tell them that you are doing a survey and politely ask them if they would mind telling you their blood type.

### Ignore the ones who refuse to answer or who don't know.

Keep going until you have 100 positive answers.
Record your results in a table, as follows:

 Blood type Tally Frequency A B AB O Total = 100

When you have finished, you can work out the relative frequency of blood type O from your sample.
Divide the number with blood type O by 100.

You can expect to get a better result if you use a bigger sample,
or if you take several samples and calculate the mean.

Now all you need to know is how many people there are in the world. Use a world population clock such as the one at http://galen.metapath.org/popclk.html. But be quick; it's changing all the time!