Explanation:
To find the weight corresponding to each event, we can use the properties of the normal distribution and the given mean and standard deviation.
Highest 30 percent:
To find the weight that corresponds to the highest 30 percent, we need to find the z-score that represents the 30th percentile. Using a standard normal distribution table or a calculator, we find that the z-score for the highest 30 percent is approximately 0.524. We can then use the z-score formula to find the corresponding weight:
z = (x - mean) / standard deviation
0.524 = (x - 355) / 12
Solving for x:
x - 355 = 0.524 * 12
x - 355 ≈ 6.29
x ≈ 361.29 grams
Therefore, the weight corresponding to the highest 30 percent is approximately 361.29 grams.
Middle 70 percent:
The middle 70 percent represents the range from the 15th percentile to the 85th percentile. Using the z-score formula, we find the z-scores for these percentiles: -1.036 and 1.036. We can then calculate the corresponding weights:
For the 15th percentile:
-1.036 = (x - 355) / 12
x - 355 = -1.036 * 12
x ≈ 342.55 grams
For the 85th percentile:
1.036 = (x - 355) / 12
x - 355 = 1.036 * 12
x ≈ 367.43 grams
Therefore, the weight corresponding to the middle 70 percent is approximately between 342.55 grams and 367.43 grams.
Highest 90 percent:
To find the weight that corresponds to the highest 90 percent, we need to find the z-score that represents the 90th percentile. The z-score is approximately 1.282. Using the z-score formula:
1.282 = (x - 355) / 12
Solving for x:
x - 355 = 1.282 * 12
x ≈ 370.18 grams
Therefore, the weight corresponding to the highest 90 percent is approximately 370.18 grams.
Lowest 20 percent:
To find the weight that corresponds to the lowest 20 percent, we need to find the z-score that represents the 20th percentile. The z-score is approximately -0.841. Using the z-score formula:
-0.841 = (x - 355) / 12
Solving for x:
x - 355 = -0.841 * 12
x ≈ 344.69 grams
Therefore, the weight corresponding to the lowest 20 percent is approximately 344.69 grams.