Assume that the weights of bananas follow a Normal distribution with a mean of 124 grams, and a standard deviation of 17 grams. Suppose that a store receives weekly shipments of 11 Bananas. The store owner will reject the shipment if the average weight is below 125g. What is the probability that the next shipment of Bananas is accepted by the store owner?Give your answer to four decimal places. If the question cannot be answered with the information given, enter -1 as your answer.
Question
Assume that the weights of bananas follow a Normal distribution with a mean of 124 grams, and a standard deviation of 17 grams. Suppose that a store receives weekly shipments of 11 Bananas. The store owner will reject the shipment if the average weight is below 125g. What is the probability that the next shipment of Bananas is accepted by the store owner?Give your answer to four decimal places. If the question cannot be answered with the information given, enter -1 as your answer.
Solution
To solve this problem, we need to use the Central Limit Theorem which states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.
Here, we are given that the population distribution is normal with a mean (μ) of 124 grams and a standard deviation (σ) of 17 grams. The sample size (n) is 11.
First, we need to calculate the mean and standard deviation of the sample distribution. The mean of the sample distribution (μx̄) is equal to the population mean (μ), which is 124 grams.
The standard deviation of the sample distribution (σx̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). So, σx̄ = σ/√n = 17/√11 ≈ 5.13 grams.
Next, we need to find the z-score for the cutoff weight of 125 grams. The z-score is calculated as (X - μx̄) / σx̄, where X is the cutoff weight. So, z = (125 - 124) /
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