Answer :
Final answer:
To calculate the probability that the mean mass of 13 pumpkins is between 4.35 kg and 4.65 kg, use the Central Limit Theorem. Find the least value of n for a probability of at least 0.90 that the mean mass is less than 4.60 kg. To find the probability that the total mass of 128 pumpkins is greater than 570 kg, use the Central Limit Theorem and assume independent and identically distributed pumpkin masses.
Explanation:
i) To calculate the probability that the mean mass of 13 pumpkins is between 4.35 kg and 4.65 kg, we can use the Central Limit Theorem. We convert the given mean and standard deviation to the mean and standard deviation of the sample mean. The mean of the sample mean is still 4.5 kg, and the standard deviation is the population standard deviation divided by the square root of the sample size, which is 0.4 kg divided by the square root of 13. We then calculate the z-scores for the lower and upper limits, and use the z-table or a calculator to find the probabilities.
ii) To find the least value of n, we need to find the z-score that corresponds to a probability of at least 0.90. We can use the z-table or a calculator to find this z-score, and then solve for the sample size n using the formula for the z-score.
iii) To find the probability that the total mass of 128 pumpkins is greater than 570 kg, we need to convert the individual pumpkin mass to the sum of the pumpkin masses. We can then use the Central Limit Theorem to approximate the distribution as normal, with the mean of 128 times the mean mass of a single pumpkin, and the standard deviation of the square root of 128 times the standard deviation of a single pumpkin. We finally use the z-table or a calculator to find the probability. The assumption made is that the pumpkin masses are independent and identically distributed.
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