Final answer:
The probability that a randomly picked fruit weighs between 518 grams and 598 grams is approximately 43.58%, calculated using the Z-scores and the standard normal distribution.
Step-by-step explanation:
To find the probability that a fruit picked at random weighs between 518 grams and 598 grams, given that the fruit's weights are normally distributed with a mean of 513 grams and a standard deviation of 38 grams, we need to use the normal distribution properties.
First, we need to calculate the Z-scores for the weights 518 grams and 598 grams:
- For 518 grams: Z = (518 - 513) / 38 = 5 / 38 ≈ 0.13
- For 598 grams: Z = (598 - 513) / 38 = 85 / 38 ≈ 2.24
Next, we look up these Z-scores in a standard normal distribution table, or use a calculator equipped with normal distribution functions, to find the probabilities corresponding to these Z-scores.
Let P1 be the probability for Z = 0.13, and P2 be the probability for Z = 2.24. The probability of the fruit weighing between 518 grams and 598 grams is P2 - P1.
Assuming P1 ≈ 0.5517 and P2 ≈ 0.9875 (from a Z-table or calculator), the probability that the fruit weighs between 518 and 598 grams is:
Probability = P2 - P1 ≈ 0.9875 - 0.5517 ≈ 0.4358 or 43.58%