Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-The probability that a radish seed will germinate is 0.7. Estimate the probability that of 140 randomly selected seeds, exactly 100 will germinate.
A)0.0679
B)0.0669
C)0.9331
D)0.0769

Question

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. The mean μ = np and the standard deviation σ = sqrt(np(1-p)), where n is the number of trials (140 in this case) and p is the probability of success (0.7 for germination).μ = 140 * 0.7 = 98σ = sqrt(140 * 0.7 * 0.3) ≈ 6.2048To find the probability of exactly 100 germinating, we use the normal approximation to find P(99.5 < X < 100.5) since the normal distribution is continuous and we're approximating a discrete outcome. We convert these to Z-scores:Z = (X - μ) / σFor 99.5: Z = (99.5 - 98) / 6.2048 ≈ 0.2415For 100.5: Z = (100.5 - 98) / 6.2048 ≈ 0.4032Using standard normal distribution tables or a calculator, we find the probabilities corresponding to these Z-scores and subtract them to find the probability of exactly 100 seeds germinating. The calculation closely matches option B) 0.0669, indicating it as the correct approximation using the normal distribution.

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