Answer :
Final answer:
Using the Central Limit Theorem, we can calculate the probabilities for various sample averages of fresh fruit consumption.
Explanation:
To calculate the probability of sample averages, we need to use the Central Limit Theorem. Given that the average annual consumption of fresh fruit per person is 99.9 pounds with a standard deviation of 30 pounds, we can use the formula: Z = (X - μ) / (σ / √n), where Z is the Z-score, X is the sample average, μ is the population mean, σ is the population standard deviation, and n is the sample size.
a. To find the probability that the sample average would be less than 90 pounds, we calculate the Z-score: Z = (90 - 99.9) / (30 / √38) = -2.41. Using a Z-table, we find that the probability is approximately 0.008, or 0.8%.
b. To find the probability that the sample average would be between 98 and 105 pounds, we calculate the Z-scores for both values: Z1 = (98 - 99.9) / (30 / √38) = -0.38 and Z2 = (105 - 99.9) / (30 / √38) = 0.78. Using a Z-table, we find the probability for Z2 and subtract the probability for Z1 to get approximately 0.353, or 35.3%.
c. To find the probability that the sample average would be less than 112 pounds, we calculate the Z-score: Z = (112 - 99.9) / (30 / √38) = 1.89. Using a Z-table, we find that the probability is approximately 0.970, or 97.0%.
d. To find the probability that the sample average would be between 93 and 96 pounds, we calculate the Z-scores for both values: Z1 = (93 - 99.9) / (30 / √38) = -1.14 and Z2 = (96 - 99.9) / (30 / √38) = -0.72. Using a Z-table, we find the probability for Z2 and subtract the probability for Z1 to get approximately 0.338, or 33.8%.
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