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We are given a normally distributed set of fruit weights with:

- Mean \( \mu = 407 \) grams
- Standard deviation \( \sigma = 21 \) grams

The problem asks us to find the weight that exceeds the heaviest 10% of fruits. This corresponds to finding the 90th percentile of the normal distribution, which can be done using the inverse cumulative distribution function (CDF).

### Steps:
1. For the 90th percentile, we need to find the corresponding z-score using the standard normal distribution. From z-tables or a calculator, we know that for 90%, the z-score \( z_{0.90} \approx 1.28 \).
   
2. Use the formula to convert the z-score to a value in the normal distribution:
   \[
   X = \mu + z \cdot \sigma
   \]
   Substituting the values:
   \[
   X = 407 + 1.28 \cdot 21
   \]
   \[
   X = 407 + 26.88 = 433.88
   \]

So, the heaviest 10% of fruits weigh more than approximately **434 grams** (rounded to the nearest gram).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the weight of the lightest 10% of fruits?
2. How would the result change if the standard deviation was 30 grams instead of 21 grams?
3. What weight corresponds to the 50th percentile (median) of this fruit distribution?
4. How can we interpret the z-score in terms of standard deviations from the mean?
5. If a fruit weighs 450 grams, what percentage of fruits weigh less than this?

**Tip**: Z-scores are useful in finding how far away a particular value is from the mean in terms of standard deviations.

A particular fruit's weights are normally distributed, with a mean of 407 grams and a standard deviation of 21 grams. The heaviest 10% of fruits weigh more than how many grams? Give your answer to the nearest gram.

Solve a statistics problem to find the weight that exceeds the heaviest 10% of fruits in a normally distributed dataset with mean 407 grams and standard deviation 21 grams. Learn how to use z-scores and the inverse cumulative distribution function. Suitable for students in Grades 10-12.

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