Answer:
To find the weight that corresponds to each event, we will use the concept of z-scores, which represent the number of standard deviations a particular value is away from the mean.
a. Lowest 15 percent:
To find the weight that corresponds to the lowest 15 percent, we need to find the z-score associated with that percentile. This z-score corresponds to the point where the cumulative probability is 0.15. Using a standard normal distribution table or functions in Excel, we can find the z-score to be approximately -1.04.
Next, we can use the formula for z-scores to find the corresponding weight:
z = (X - mean) / standard deviation
-1.04 = (X - 415) / 23
Solving for X, we get:
X = -1.04 * 23 + 415 ≈ 391.88 grams
Therefore, the weight corresponding to the lowest 15 percent is approximately 391.88 grams.
b. Highest 20 percent:
Similar to the previous case, we need to find the z-score associated with the highest 20 percent. This z-score corresponds to the point where the cumulative probability is 0.80 (1 - 0.20). Using a standard normal distribution table or functions in Excel, we find the z-score to be approximately 0.84.
Again, using the formula for z-scores:
0.84 = (X - 415) / 23
Solving for X, we get:
X = 0.84 * 23 + 415 ≈ 434.92 grams
Therefore, the weight corresponding to the highest 20 percent is approximately 434.92 grams.
c. Middle 60 percent:
The middle 60 percent corresponds to the range between the 20th and 80th percentiles. To find the z-scores associated with these percentiles, we can subtract and add half the range from the mean.
Lower z-score:
20th percentile = 0.20
Lower z-score = (1 - 0.60) / 2 = -0.20
Using the formula for z-scores:
-0.20 = (X - 415) / 23
Solving for X, we get:
X = -0.20 * 23 + 415 ≈ 410.60 grams
Upper z-score:
80th percentile = 0.80
Upper z-score = (1 + 0.60) / 2 = 0.80
Using the formula for z-scores:
0.80 = (X - 415) / 23
Solving for X, we get:
X = 0.80 * 23 + 415 ≈ 419.40 grams
Therefore, the weight corresponding to the middle 60 percent ranges approximately from 410.60 grams to 419.40 grams.
d. Highest 80 percent:
To find the weight that corresponds to the highest 80 percent, we can subtract half the remaining range from the mean.
Remaining range:
100% - 80% = 20%
Half of the remaining range:
20% / 2 = 10%
Lower z-score:
80th percentile = 0.80
Lower z-score = (1 - 0.10) = 0.90
Using the formula for z-scores:
0.90 = (X - 415) / 23
Solving for X, we get:
X = 0.90 * 23 + 415 ≈ 432.70 grams
Therefore, the weight corresponding to the highest 80 percent is approximately 432.70 grams.
In summary:
a. Lowest 15 percent: Approximately 391.88 grams.
b. Highest 20 percent: Approximately 434.92 grams.
c. Middle 60 percent: Approximately ranging from 410.60 grams to 419.40 grams.
d. Highest 80 percent: Approximately 432.70 grams.