A vegetable distributor knows that during the month of August, the weights of tomatoes are normally distributed with a mean of 0.61 lb and a standard deviation of 0.15 lb. How many tomatoes can be expected to weigh between 0.31 lb and 0.91 lb in a shipment of 4,500 tomatoes?

Select one:
A. 4,000
B. 4,275
C. 4,100
D. 4,215

To find the number of tomatoes that can be expected to weigh between 0.31 to 0.91 lbs in a shipment of 4500 tomatoes, we use the z-score formula and the z-table to calculate probabilities. The expected number of tomatoes is 4275.

Explanation:

To find the number of tomatoes that can be expected to weigh between 0.31 to 0.91 lbs in a shipment of 4500 tomatoes, we need to calculate the z-scores for both weights and then use the z-table to find the probabilities. The z-score formula is z = (x - μ) / σ, where x is the weight, μ is the mean, and σ is the standard deviation. For 0.31 lbs: z = (0.31 - 0.61) / 0.15 = -2.00, and for 0.91 lbs: z = (0.91 - 0.61) / 0.15 = 2.00.

Next, we use the z-table to find the probabilities for the z-scores. From the z-table, the probability of a z-score less than -2.00 is 0.0228, and the probability of a z-score less than 2.00 is 0.9772. So, the probability of a tomato weighing between 0.31 to 0.91 lbs is 0.9772 - 0.0228 = 0.9544.

Finally, we multiply the probability by the total number of tomatoes to find the expected number of tomatoes within that weight range. Expected count = 0.9544 * 4500 = 4279.8. Since the count must be a whole number, the expected number of tomatoes that can be expected to weigh between 0.31 to 0.91 lbs in a shipment of 4500 tomatoes is 4275.

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