Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-Estimate the probability of getting exactly 43 boys in 90 births.
A)0.0159
B)0.0729
C)0.1628
D)0.0764

Question

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-Estimate the probability of getting exactly 43 boys in 90 births. 
A)0.0159
B)0.0729
C)0.1628
D)0.0764

Quizplus · Accepted Answer

The normal approximation to the binomial distribution can be used here with the mean μ = np and the standard deviation σ = sqrt(np(1-p)), where n is the number of trials (births, in this case, 90) and p is the probability of a success (having a boy, assumed to be 0.5 for simplicity). Thus, μ = 90*0.5 = 45 and σ = sqrt(90*0.5*0.5) = sqrt(22.5) ≈ 4.7434. To find the probability of getting exactly 43 boys, we use the continuity correction and look for the probability of getting a result between 42.5 and 43.5. Using the Z-score formula Z = (X - μ) / σ, we find the Z-scores for 42.5 and 43.5, and then use the standard normal distribution to find the probabilities corresponding to these Z-scores. The difference between these probabilities gives an approximation of the probability of getting exactly 43 boys. The closest value to the calculated probability using this method is option D) 0.0764.

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