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Question: A farmer finds the mean mass for a random sample of 200 eggs laid by his hens to be 57.2 grams. If the masses of eggs for this breed of hen are normally distributed with standard deviation 1.5 grams, estimate the mean mass, to the nearest tenth of a gram, of the eggs for this breed using a 99% confidence interval.
To estimate the mean mass of the eggs for this breed using a 99% confidence interval, we will use the formula for the confidence interval of a mean with a known standard deviation: CI = x̄ ± (z * (σ / √n)) where: - CI is the confidence interval - x̄ is the sample mean (57.2 grams) - z is the z-score associated with the desired confidence level (99%) - σ is the known standard deviation (1.5 grams) - n is the sample size (200 eggs) First, we find the z-score for a 99% confidence interval. You can look up the z-score in a z-table, or use an online calculator. The z-value for a 99% confidence interval (which leaves 1% in the tails) is 2.576 (rounded to three decimal places). Now we can plug in the values into the formula: CI = 57.2 ± (2.576 * (1.5 / √200)) Now, calculate within the parentheses: CI = 57.2 ± (2.576 * (1.5 / 14.142)) CI = 57.2 ± (2.576 * 0.106) Now, multiply and add/subtract: CI = 57.2 ± 0.273 So, our 99% confidence interval for the mean mass of the eggs for this breed of hen is approximately (56.9, 57.5) grams, to the nearest tenth of a gram. This means we are 99% confident that the true mean mass of the eggs for this breed of hen falls within this interval.
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