Question

Random samples of size n=80 were selected from a binomial population with n=0.3 p=0.3. Use the normal distribution to approximate the following probability. (Round your answer to four decimal places.)

a) 0.0399
b) 0.9614
c) 0.0386
d) 0.9627

Answer 1

To approximate binomial probabilities using the normal distribution for n=80 and p=0.3, firstly verify np and nq are greater than five. The mean is calculated as μ=np and standard deviation as σ=√npq. Next, find the z-score for the specific value and use the standard normal distribution to approximate the probabilities.

Step-by-step explanation:

To use the normal distribution to approximate the probability for a binomial population where n=80 and p=0.3, we need to check if the approximation conditions are met; specifically, that np and nq (where q=1-p) are both greater than five. Here, np=80(0.3)=24 and nq=80(0.7)=56, so the conditions are satisfied. We then calculate the mean (μ) as np (which is 24) and the standard deviation (σ) as √npq, which is √(80)(0.3)(0.7) approx 4.3818. To find the associated z-scores, we need to adjust our x value by ±0.5, then use z=(x-μ)/σ. To approximate the probabilities, utilize the standard normal distribution table or a calculator's normal distribution function.