Question

5:17 PM Sun Nov 10 46% moodle31.upei.ca Consider the stock returns/lottery example with four possible out- 3 comes I discussed in class. (I used Red, Blue, Yellow and Green little balls to represent the outcomes.) Now assume that only three outcomes are possible with the following probability distribution: % Rate of Return (x) p(r) Stock 1/3 Red 1 1/3 Blue 2 1/3 Yellow 3 1. You can imagine, for example, that there is a stock index with 300 stocks, 100 being Red, 100 Blue and 100 Yellow Suppose you take a sample of n 2 and consider all possible samples of n = 2. Find the distribution of the sample mean X and the corresponding ux and ag values. Suppose you take a sample of n = 3 and consider all possible samples of n = 3. Find the distribution of the sample mcan X and the corresponding uz and a^ values. For this part you can refer to the Excel file <Q4(b)-Possible Samples-n3.xlsx> for a list of all possible samples of n 3 What is your conclusion regarding the shape of the distributions of X and ug and og values?

Answer 1

1)

n= 2

x1	x2	Xbar
1	1	1
1	2	1.5
1	3	2
2	1	1.5
2	2	2
2	3	2.5
3	1	2
3	2	2.5
3	3	3
	mean	2
	sd_xbar	0.612372

Formula

x1	x2	Xbar
1	1	=(A2+B2)/2
=A2	2	=(A3+B3)/2
=A3	3	=(A4+B4)/2
2	1	=(A5+B5)/2
=A5	2	=(A6+B6)/2
=A6	3	=(A7+B7)/2
3	1	=(A8+B8)/2
=A8	2	=(A9+B9)/2
=A9	3	=(A10+B10)/2
	mean	=AVERAGE(C2:C10)
	sd_xbar	=STDEV(C2:C10)

2)

n = 3

x1	x2	x3	xbar
1	1	1	1
1	1	2	1.333333
1	1	3	1.666667
1	2	1	1.333333
1	2	2	1.666667
1	2	3	2
1	3	1	1.666667
1	3	2	2
1	3	3	2.333333
2	1	1	1.333333
2	1	2	1.666667
2	1	3	2
2	2	1	1.666667
2	2	2	2
2	2	3	2.333333
2	3	1	2
2	3	2	2.333333
2	3	3	2.666667
3	1	1	1.666667
3	1	2	2
3	1	3	2.333333
3	2	1	2
3	2	2	2.333333
3	2	3	2.666667
3	3	1	2.333333
3	3	2	2.666667
3	3	3	3
		mean	2
		sd_Xbar	0.480384

3)

as sample size increases , sample mean follows normal distribution

mu_xbar remain same as mu

sigma_xbar = sigma/sqrt(n)

hence as n increases sigma_xbar decreases

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