IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem
X follows a Binomial(100, 0.5) distribution. Using a normal approximation with continuity correction, what is the approximate value of P(X ≤ 55)?
The answer is 0.8643. With mean 50 and standard deviation 5, the discrete event X ≤ 55 is treated as the continuous event Y < 55.5. This gives z = 1.1, and the standard normal probability below 1.1 is 0.8643.
- A0.8159
- B0.8413
- C0.8643Correct
- D0.9032
- 0.9332
Explanation
The mean is 50 and the standard deviation is 5. P(X ≤ 55) becomes P(Y < 55.5), so z = 5.5/5 = 1.1 and Φ(1.1) = 0.8643. Omitting the correction gives z = 1 and 0.8413. Using 54.5 gives 0.8159, which is the wrong direction.
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