Math 2810 Course Notes 2/28
4.11. The Central Limit Theorem
Theorem 1. (Central Limit Theorem) If X1, X2, . . . , Xnis a large (n > 30) sample from a
population with mean µand variance σ2, then
1. (Sum) Sn≈N(nµ, nσ2), where Sn=X1+X2+· ·· +Xn.
2. (Average) X=Sn
n≈Nµ, σ2
n.
Example 2. Suppose ages of students in a university have population mean 22.3 years and
standard deviation 4 years. Estimate the probability that the average age of 64 randomly
selected students is greater than 23 years old.
Let Xbe the average age of the 64 students. According to the Central Limit Theorem,
X≈N22.3,42
64=N(22.3,(0.5)2).
So
P(X > 23) = PZ > 23 −22.3
0.5=P(Z > 1.4) = 1 −Φ(1.4).
Example 3. Estimate the probability that the sum of 100 fair die rolls is greater than 400.
Recall one dice roll has population mean µ=7
2= 3.5 and variance σ2=35
12 ≈2.9167.
Let S100 be the sum of 100 dice rolls. According to the Central Limit Theorem,
S100 ≈N(100 ·3.5,100 ·2.9167) = N(350,291.67).
Therefore
P(S100 >400) = PZ > 400 −350
√291.67 ≈P(Z > 2.93) = 1 −Φ(2.93).
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