Question

One hundred draws will be made at random with replacement from the box with the following numbers: 1 6 7 9 9 10 What is the expected value of a sum of 100 draws from this box?

Using the same box of numbers described above, what is the standard error of a sum of 100 draws from this box?

Using the same box of numbers described above, what is the chance of getting a sum between 650 and 750?

Answer 1

E(Xi) = 1/6 ( 1+ 6 +7 + 9 + 9 +10) = 7

x	p	px	px^2
1	0.166666667	0.166666667	0.166667
6	0.166666667	1	6
7	0.166666667	1.166666667	8.166667
9	0.333333333	3	27
10	0.166666667	1.666666667	16.66667
	1	7	58

Var(Xi) = E(X^2) - E(X) ^2

= 58 - 7^2 = 58 -49 = 9

What is the expected value of a sum of 100 draws from this box?

S = X1+X2+..X100

E(S) = 100 E(X) = 700

Using the same box of numbers described above, what is the standard error of a sum of 100 draws from this box?

Var(S) = 100 * 9 = 900

Using the same box of numbers described above, what is the chance of getting a sum between 650 and 750?

since n= 100 > 30

we can use central limit theorem

Z = (S - 700)/sqrt(900) = (S - 700)/30

P( 650 <S< 750)

= 0.9044