Explanation:
To find the weight that corresponds to each event, we need to use the z-score formula and standard normal distribution tables. The z-score formula is given by:
z = (x - μ) / σ
Where:
z = z-score
x = observed value
μ = mean
σ = standard deviation
a. Highest 30 percent:
To find the weight corresponding to the highest 30 percent, we need to find the z-score that corresponds to a cumulative probability of 0.30 from the standard normal distribution table.
Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52.
Now we can calculate the weight:
z = (x - μ) / σ
-0.52 = (x - 340) / 11
Solving for x:
x - 340 = -0.52 * 11
x - 340 = -5.72
x = 334.28
Therefore, the weight corresponding to the highest 30 percent is approximately 334.28 grams.
b. Middle 70 percent:
To find the weight corresponding to the middle 70 percent, we need to find the z-scores that correspond to the lower and upper percentiles. The lower percentile would be (100 - 70) / 2 = 15 percent, and the upper percentile would be 100 - 15 = 85 percent.
Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.15 is approximately -1.04, and the z-score corresponding to a cumulative probability of 0.85 is approximately 1.04.
Now we can calculate the weights:
Lower weight:
z = (x - μ) / σ
-1.04 = (x - 340) / 11
x - 340 = -1.04 * 11
x - 340 = -11.44
x = 328.56
Upper weight:
z = (x - μ) / σ
1.04 = (x - 340) / 11
x - 340 = 1.04 * 11
x - 340 = 11.44
x = 351.44
Therefore, the weight corresponding to the middle 70 percent is approximately between 328.56 grams and 351.44 grams.
c. Highest 90 percent:
To find the weight corresponding to the highest 90 percent, we need to find the z-score that corresponds to a cumulative probability of 0.90 from the standard normal distribution table.
Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.90 is approximately 1.28.
Now we can calculate the weight:
z = (x - μ) / σ
1.28 = (x - 340) / 11
Solving for x:
x - 340 = 1.28 * 11
x - 340 = 14.08
x = 354.08
Therefore, the weight corresponding to the highest 90 percent is approximately 354.08 grams.
d. Lowest 20 percent:
To find the weight corresponding to the lowest 20 percent, we need to find the z-score that corresponds to a cumulative probability of 0.20 from the standard normal distribution table.
Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.20 is approximately -0.84.
Now we can calculate the weight:
z = (x - μ) / σ
-0.84 = (x - 340) / 11
Solving for x:
x - 340 = -0.84 *
11
x - 340 = -9.24
x = 330.76
Therefore, the weight corresponding to the lowest 20 percent is approximately 330.76 grams.