Question

A vegetable distributor knows that during the month of August, the weights of its tomatoes are normally distributed with a mean of 0.54 lb and a standard deviation of 0.15 lb. (See Example 2 in this section.)

(a) What percent of the tomatoes weigh less than 0.69 lb?

(b) In a shipment of 8,000 tomatoes, how many tomatoes can be expected to weigh more than 0.24 lb?

(c) In a shipment of 4,500 tomatoes, how many tomatoes can be expected to weigh from 0.24 lb to 0.84 lb?

Answer 1

The percent of tomatoes weighing less than 0.69 lb is approximately 84.13%, in a shipment of 8,000 tomatoes, approximately 183.36 tomatoes are expected to weigh more than 0.24 lb, and in a shipment of 4,500 tomatoes, approximately 4,294.8 tomatoes are expected to weigh from 0.24 lb to 0.84 lb.

Step-by-step explanation:

(a) To find the percent of tomatoes that weigh less than 0.69 lb, we need to calculate the z-score and then use the z-table to find the corresponding percentile. The z-score is calculated using the formula: z = (X - mean) / standard deviation. Substituting the given values, we get z = (0.69 - 0.54) / 0.15 = 1. The percentile associated with a z-score of 1 is approximately 84.13%. Therefore, approximately 84.13% of the tomatoes weigh less than 0.69 lb.

(b) To find the number of tomatoes that can be expected to weigh more than 0.24 lb in a shipment of 8,000 tomatoes, we again calculate the z-score using the formula: z = (X - mean) / standard deviation. Substituting the given values, we get z = (0.24 - 0.54) / 0.15 = -2. The area to the left of a z-score of -2 is approximately 0.0228. Therefore, there are approximately 0.0228 * 8,000 = 183.36 tomatoes expected to weigh more than 0.24 lb.

(c) To find the number of tomatoes that can be expected to weigh from 0.24 lb to 0.84 lb in a shipment of 4,500 tomatoes, we calculate the z-scores for 0.24 lb and 0.84 lb. The z-score for 0.24 lb is (0.24 - 0.54) / 0.15 = -2, and the z-score for 0.84 lb is (0.84 - 0.54) / 0.15 = 2. The area between these two z-scores is approximately 0.9772 - 0.0228 = 0.9544. Therefore, there are approximately 0.9544 * 4,500 = 4,294.8 tomatoes expected to weigh from 0.24 lb to 0.84 lb.

Answer 2

To find the percent of tomatoes that weigh less than 0.69 lb, calculate the z-score and use the z-table. For the shipment of 8,000 tomatoes, calculate the z-score for 0.24 lb and find the percentile. For the shipment of 4,500 tomatoes, calculate the z-scores for both weights and find the percentiles.

Step-by-step explanation:

To find the percent of tomatoes that weigh less than 0.69 lb, we need to calculate the z-score for that weight and use the z-table to find the corresponding percentile. The formula for calculating the z-score is: z = (x - mean) / standard deviation. Substituting the values, we get: z = (0.69 - 0.54) / 0.15 = 1.

Using the z-table, we find that the percentile corresponding to a z-score of 1 is 0.8413. So, approximately 84.13% of the tomatoes weigh less than 0.69 lb.

To find the number of tomatoes that can be expected to weigh more than 0.24 lb in a shipment of 8,000 tomatoes, we calculate the z-score for 0.24 lb and use the z-table to find the corresponding percentile. Using the formula z = (x - mean) / standard deviation, we get: z = (0.24 - 0.54) / 0.15 = -2.

Looking up the z-score -2 in the z-table, we find that the percentile is 0.0228. So, approximately 0.0228 * 8000 = 182.4 tomatoes can be expected to weigh more than 0.24 lb.

To find the number of tomatoes that can be expected to weigh from 0.24 lb to 0.84 lb in a shipment of 4,500 tomatoes, we calculate the z-scores for both weights and use the z-table to find the percentiles. First, we calculate the z-score for 0.24 lb: z = (0.24 - 0.54) / 0.15 = -2. Then, we calculate the z-score for 0.84 lb: z = (0.84 - 0.54) / 0.15 = 2.

Using the z-table, we find that the percentile corresponding to a z-score of -2 is 0.0228, and the percentile corresponding to a z-score of 2 is 0.9772. So, the percentage of tomatoes weighing from 0.24 lb to 0.84 lb is approximately (0.9772 - 0.0228) * 4500 = 4463.2 tomatoes.