To find the weight above which the heaviest 8% of fruits fall, we can use the properties of the normal distribution.
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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%).
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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.
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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).
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Plug in the values: \[ X = 415 + (1.41 \times 18) \] \[ X = 415 + 25.38 \] \[ X \approx 440.38 \]
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Rounding to the nearest gram gives us: \[ X \approx 440 \text{ grams} \]
Thus, the heaviest 8% of fruits weigh more than 440 grams.