To find the probability that a randomly chosen fruit will weigh between 609 grams and 659 grams, we need to standardize the values using the Z-score formula for normal distribution:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X$ is the weight we are interested in,
• $\mu$ is the mean (639 grams),
• $\sigma$ is the standard deviation (14 grams).

Step 1: Calculate the Z-scores

For $X = 609$: $Z = \frac{609 - 639}{14} = \frac{-30}{14} \approx -2.14$

For $X = 659$: $Z = \frac{659 - 639}{14} = \frac{20}{14} \approx 1.43$

Step 2: Use Z-tables or a standard normal distribution calculator

Now, we look up the probabilities for these Z-scores:

• $P(Z = -2.14) \approx 0.0162$ (this is the cumulative probability up to 609 grams),
• $P(Z = 1.43) \approx 0.9236$ (this is the cumulative probability up to 659 grams).

Step 3: Find the probability between 609 and 659 grams

The probability that the fruit weighs between 609 and 659 grams is the difference between these two probabilities:

$P(609 < X < 659) = P(Z = 1.43) - P(Z = -2.14) = 0.9236 - 0.0162 = 0.9074$

So, the probability is approximately 0.9074 or 90.74%.

Would you like more details or have any questions?

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Here are 5 related questions you might explore:

1. What is the probability that the fruit weighs more than 659 grams?
2. What is the probability that the fruit weighs less than 609 grams?
3. How do you calculate probabilities for values further from the mean?
4. How would changing the standard deviation to 20 grams affect the probability?
5. What would the Z-score be for a fruit that weighs 680 grams?

Tip: When calculating Z-scores, always ensure the values are based on the same units (such as grams) as the mean and standard deviation.