Answer :
Final answer:
a) The probability the mean incubation time will last 20 days is approximately 15.87%. b) The incubation times that make up the middle 95% of fertilized chicken eggs are between μ + (-1.96) * (σ / √n) and μ + (1.96) * (σ / √n).
Explanation:
a) To find the probability that the mean incubation time will last 20 days, we need to calculate the z-score and use the z-table. The formula for the z-score is: z = (x - μ) / (σ / √n), where x is the value we are interested in, μ is the mean, σ is the standard deviation, and n is the sample size. Plugging in the values, we get: z = (20 - 21) / (1 / √1) = -1. Using the z-table, we can find that the probability associated with a z-score of -1 is approximately 0.1587, which is equivalent to 15.87%.
b) To determine the incubation times that make up the middle 95% of fertilized chicken eggs, we need to find the z-scores corresponding to the lower and upper percentiles. The lower percentile is (100 - 95) / 2 = 2.5%, and the upper percentile is 100 - 2.5 = 97.5%. Using the z-table, we can find that the z-score corresponding to a lower percentile of 2.5% is approximately -1.96, and the z-score corresponding to an upper percentile of 97.5% is approximately 1.96. We can then plug these z-scores back into the z-score formula to find the corresponding incubation times: lower time = μ + (-1.96) * (σ / √n) and upper time = μ + (1.96) * (σ / √n).
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