Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-With n = 20 and p = 0.60, estimate P(less than or equal to 8).
A)0.4953
B)0.4332
C)0.0548
D)0.0668

Question

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-With n = 20 and p = 0.60, estimate P(less than or equal to 8). 
A)0.4953
B)0.4332
C)0.0548
D)0.0668

Quizplus · Accepted Answer

To estimate the probability using the normal distribution as an approximation to the binomial distribution, we first calculate the mean (μ) and standard deviation (σ) of the binomial distribution. For a binomial distribution, μ = np and σ = sqrt(np(1-p)). Here, n = 20 and p = 0.60, so μ = 20*0.60 = 12 and σ = sqrt(20*0.60*0.40) = sqrt(4.8) ≈ 2.19. To find P(X ≤ 8), we convert the binomial variable X to a standard normal variable Z using the formula Z = (X - μ) / σ. For X = 8, Z = (8 - 12) / 2.19 ≈ -1.83. Looking up this Z-value in a standard normal distribution table or using a calculator gives a probability of approximately 0.0336. However, because we are using the normal approximation, we apply the continuity correction factor by adjusting our X value to 8.5 (since we're looking for P(X ≤ 8), we consider P(X < 8.5) for the normal approximation). Recalculating Z with X = 8.5 gives Z ≈ (8.5 - 12) / 2.19 ≈ -1.60. Looking up this corrected Z-value gives a probability closer to the options provided, and among the choices, 0.0548 is the closest to the correct approximation, acknowledging some discrepancy due to rounding or different continuity correction interpretations.

Estimate the Indicated Probability by Using the Normal Distribution as an Approximation