3. Four hundred tickets are drawn at random with replacement from the box [0,0,0,1,2,3]
a) What is the expected value of the sum of the draws?
b) What is the standard error for the sum of the draws?
c) What is the probability that the average of the draws is more than 1.075?
here from above let X is random number drawn in 1 pick up.
x | P(x) | xP(x) | x^{2}P(x) |
0 | 1/2 | 0.000 | 0.000 |
1 | 1/6 | 0.167 | 0.167 |
2 | 1/6 | 0.333 | 0.667 |
3 | 1/6 | 0.500 | 1.500 |
total | 1.000 | 2.333 | |
E(x) =μ= | ΣxP(x) = | 1.0000 | |
E(x^{2}) = | Σx^{2}P(x) = | 2.3333 | |
Var(x)=σ^{2} = | E(x^{2})-(E(x))^{2}= | 1.333 | |
std deviation= | σ= √σ^{2} = | 1.1547 |
a) expected value of the sum of 400 draws =400*1=400
b) standard error for the sum of the draws =1.1547*sqrt(400)=23.094
c)
from normal approximation;
probability that the average of the draws is more than 1.075
=P(expected sum is greater then 1.075*400=430)=P(X>430)=P(Z>(430-400)/23.094)
=P(Z>1.30)=0.0968
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