Question

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

How many different samples of 3 pennies out of 25 can you get?

We need to choose 3 pennies out of 25, this is

ans: 2300

X be the number of marked coins out of 3 selected from 25 coins, X can take the values 0,1,2,3

If X=x is the number of marked coins in 3, then 3-x is the number of unmarked coins in 3

We need to select x coins out of 10 marked coins and 3-x coins from 15 unmarked coins

The total number of ways of doing this is

Now we will calculate this for X=0,1,2,3

ans: 675

The probabilities are

The expected value of X is

The expectation of is

The variance of X is

The standard deviation of X is

ans: the mean of X is 1.2 and the standard deviation is 0.8124

If you perform the experiment as described, one set of numbers could be

2 1 0 0 0 2 1 2 1 1 1 0 2 2 1 2 1 3 2 1

The sample proportions are

X	Frequency	Proportion
0	4	0.2
1	8	0.4
2	7	0.35
3	1	0.05
Total	20	1

The sample mean is

The sample standard deviation is

The expected mean is 1.2 and the sample mean is 1.25, the expected standard deviation is 0.8124 and the sample standard deviation is 0.8507.

They are close enough as the sample mean and sample variance are the unbiased estimators of the population mean and variance

The sample estimates will get closer as the sample size (currently at 20) increases.