Question

f 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

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(ii) Experimental data :

Nos	A	B	C	# of M
1	U	U	M	1
2	U	M	U	1
3	M	U	U	1
4	U	U	M	1
5	U	U	U	0
6	U	U	M	1
7	U	U	U	0
8	M	U	U	1
9	U	U	U	0
10	U	M	U	1
11	U	M	U	1
12	U	M	U	1
13	U	U	U	0
14	U	M	M	2
15	M	M	U	2
16	U	U	M	1
17	M	U	M	2
18	M	M	U	2
19	M	U	M	2
20	U	U	M	1
Nos	A	B	C	# of M
1	U	U	M	1
2	U	M	U	1
3	M	U	U	1
4	U	U	M	1
5	U	U	U	0
6	U	U	M	1
7	U	U	U	0
8	M	U	U	1
9	U	U	U	0
10	U	M	U	1
11	U	M	U	1
12	U	M	U	1
13	U	U	U	0
14	U	M	M	2
15	M	M	U	2
16	U	U	M	1
17	M	U	M	2
18	M	M	U	2
19	M	U	M	2
20	U	U	M	1

Calculations

X	#	P(X)	X*P(X)	X^2	x^2 * P(X)
0	4	0.2	0	0	0
1	11	0.55	0.55	1	0.55
2	5	0.25	0.5	4	1
3	0	0	0	9	0
	20		1.05		1.55

Yes the mean and std dev of random variable and data set are equal.

Sorry dont know why. If you get to know please comment.