Question

A particular fruit's weights are normally distributed, with a mean of 415 grams and a standard deviation of 18 grams.

The heaviest 8% of fruits weigh more than how many grams?

Give your answer to the nearest gram.

Answer 1

To find the weight above which the heaviest 8% of fruits fall, we can use the properties of the normal distribution.

1. **Identify the z-score** for the top 8%. Since we are looking for the heaviest 8%, we want the 92nd percentile of the normal distribution (100% - 8% = 92%).

2. **Use a z-table or statistical software** to find the z-score corresponding to the 92nd percentile. The z-score for the 92nd percentile is approximately 1.41.

3. **Use the formula for converting z-scores to raw scores**:
\[
X = \mu + z \times \sigma
\]
where:
- \( \mu = 415 \) grams (mean),
- \( z = 1.41 \) (z-score for 92nd percentile),
- \( \sigma = 18 \) grams (standard deviation).

4. **Plug in the values**:
\[
X = 415 + (1.41 \times 18)
\]
\[
X = 415 + 25.38
\]
\[
X \approx 440.38
\]

5. **Rounding to the nearest gram** gives us:
\[
X \approx 440 \text{ grams}
\]

Thus, the heaviest 8% of fruits weigh more than **440 grams**.