Question

-This is a hands-on exercise in probability distributions and sampling. To complete this problem, you will need 25 pennies and a marker. Take 10 pennies and label them 1 through 10, the same number on both sides. If you don’t want to permanently mark your pennies, use a washable marker or put some tape on the 10 marked pennies and write on the tape.

Part A: Human Beings Aren’t Good at Being Random Think about flipping a fair coin 100 times and recording H for heads or T for tails as you go along. You’d end up with 100 random Hs & Ts in a string like:

TTH.....................................................T (first was T, second was T, third was H...100th was T)

Because you have studied statistics you expect roughly 50 Hs—maybe 47 or 51—but probably not 94 Hs and 6 Ts.

I want you to do your best to write out such a random string; you are mentally flipping a coin 100 times in a row. It should take you just a minute or two. Don’t flip any coins, just imagine it. Do this before you read any further.

Now I’d like you to really flip a penny 100 times, recording H or T in order. Well, this would be a pain— you could do this with a penny, but it would waste a lot of your time. We can effectively do the same thing by putting your 10 marked pennies in a jar, shaking them, and pouring them out. Then record, in order, the ten H/Ts that show up. I did this in the picture above, and wrote down THTHHHTHHT because the coins follow the pattern below:

#              1   2     3    4     5 6     7     8     9     10

showing    T   H   T   H   H   H    T    H    H      T

Do this 10 times and write down the resulting 100 H/Ts in order, so you have 100 letters in a row. Now you have two strings of 100 H/Ts each. Email those strings to your instructor. Your instructor will email back to you the rest of Part A after reviewing your strings. (I bet the suspense is killing you).

Part B: Sampling and Random Variable You already have ten marked pennies (ones with numbers from Part A) and 15 unmarked pennies.

Thought experiment: Throw them all in a jar and shake. Without looking, pull three out and record how many of them are marked (have a number). You will get 0, 1, 2, or 3 marked coins.

How many different samples of 3 pennies out of 25 can you get? (Order doesn’t matter.) Answer: 2,300                      Show why 2,300 is the answer.

How many of those samples would have 0, 1, 2, 3 marked coins?

# of marked pennies in sample          0                         455          1                         1,050          2                          675           (Show why this count should be 675.)          3                          120                       Total     2,300       (Notice this total matches the total above.)

If you draw a simple random sample of size 3, each sample is equally likely. Counting the number of marked coins gives us a discrete random variable, X.

P(X=0)=      455/2,300     =    .1978 P(X=1)= P(X=2)= P(X=3)=

Find the rest of these probabilities. Then find the mean and standard deviation of this discrete random variable.

Now, really and truly put the ten marked coins in a jar with 15 unmarked ones. Shake, pull out three without looking. Write down the number of marked coins. Put them all back, shake, draw again, and count marked coins again. Do this a total of 20 times. Now you have 20 pieces of data. Write down the data set. Compute the sample proportions. How do your sample proportions compare to the probabilities you computed above? Find the mean and standard deviation of the data set. Are the expected value and standard deviation of the random variable close to the mean and standard deviation of the data set? Should they be? Why?

Answer 1