Final answer:
By approximating the binomial distribution with a normal distribution, the probability that at least 34 out of 65 lambs will be female is calculated to be approximately 40.2%, making the correct answer B. 40.2%.
Step-by-step explanation:
To calculate the probability that out of 65 lambs born on Wooly Farm, at least 34 will be female, we need to use binomial probability. Since the probability of a lamb being female is 0.5 (given that males and females are equally probable), we need to sum up the probabilities from 34 to 65 females.
However, manually calculating this would be very tedious and impractical. Instead, we can use a normal approximation to the binomial distribution because the number of trials (n=65) is large enough.
First, calculate the mean (μ) and standard deviation (σ) of the binomial distribution:
- μ = np = 65 × 0.5 = 32.5
- σ = √(np(1-p)) = √(65 × 0.5 × 0.5) ≈ 4.03
Since we're finding the probability for 'at least 34', we need to find the z-score for 33.5 (to include everything greater than 34) and then subtract this probability from 1.
Z = (33.5 - μ) / σ = (33.5 - 32.5) / 4.03 ≈ 0.248
Consulting a standard normal distribution table or using a calculator for the above z-score, we would find the probability to the left of the z-score. The complement of this probability will give us the probability for 'at least 34'.
Assuming we consult a standard normal distribution table and find that the z-score of 0.248 corresponds to a probability of approximately 0.5982 (59.82%), we subtract this value from 1 to find the probability of at least 34 female lambs.
1 - 0.5982 = 0.4018 or 40.18%, which rounds to 40.2%.
Therefore, the correct answer is B. 40.2%.