Question 1185702: The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 370 grams and a standard deviation of 15 grams. Find the weight that corresponds to each event. (Use Excel or Appendix C to calculate the z-value. Round your final answers to 2 decimal places.)
Highest 30 percent
Middle 70 percent
Highest 90 percent
Lowest 20 percent
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the weights corresponding to each event, using z-scores and the properties of a normal distribution:
**1. Highest 30 Percent:**
* We want to find the weight *x* such that P(X > x) = 0.30. This means that 70% of the weights are *below* this value.
* Find the z-score corresponding to a cumulative probability of 0.70. Using a z-table or calculator, we find that z ≈ 0.52.
* Use the z-score formula to find the weight:
x = μ + zσ
x = 370 + (0.52 * 15)
x = 370 + 7.8
x ≈ 377.80 grams
**2. Middle 70 Percent:**
* If the middle 70% is included, that leaves 30% in the tails, or 15% in *each* tail.
* To find the lower bound, find the z-score corresponding to a cumulative probability of 0.15 (15%). z ≈ -1.04.
* To find the upper bound, find the z-score corresponding to a cumulative probability of 0.85 (15% in each tail, so 100% - 15% = 85%). z ≈ 1.04.
Now, calculate the weights:
* Lower bound: x = 370 + (-1.04 * 15) = 370 - 15.6 ≈ 354.40 grams
* Upper bound: x = 370 + (1.04 * 15) = 370 + 15.6 ≈ 385.60 grams
**3. Highest 90 Percent:**
* If 90% of the weights are below this value, then we're looking for the weight such that P(X > x) = 0.10, so 90% of the weights are *below* this value.
* Find the z-score corresponding to a cumulative probability of 0.90. z ≈ 1.28
* Calculate the weight:
x = 370 + (1.28 * 15)
x = 370 + 19.2
x ≈ 389.20 grams
**4. Lowest 20 Percent:**
* We want to find the weight *x* such that P(X < x) = 0.20.
* Find the z-score corresponding to a cumulative probability of 0.20. z ≈ -0.84
* Calculate the weight:
x = 370 + (-0.84 * 15)
x = 370 - 12.6
x ≈ 357.40 grams
**Final Answers (rounded to 2 decimal places):**
* Highest 30 percent: 377.80 grams
* Middle 70 percent: Between 354.40 and 385.60 grams
* Highest 90 percent: 389.20 grams
* Lowest 20 percent: 357.40 grams
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