Question 1192227: A certain industrial material is packed in 2 kilogram packets. Packets selected at random have a mean mass of mean mass of 2025 with a standard deviation of 15 grams. If the masses are normally distributed, determine the percentage of packets having a mass
i. less than 2 kilograms
ii. More than 2030 grams
iii. The manufacturer decides to allow 10% of the packets to be sold underweight. If he is able to maintain the same standard deviation, what mass should he adjust his mean?
iv. A government inspector allows this but insists that only 10 % of the packets may be less than 1995 grams. To what must the manufacturer now adjust his standard deviation to meet this new requirement, if the mean is not altered?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**i. Packets less than 2 kilograms (2000 grams):**
1. **Calculate the z-score:**
z = (x - μ) / σ
z = (2000 - 2025) / 15
z = -1.67
2. **Find the probability:**
Using a z-table or calculator, the probability of a z-score less than -1.67 is approximately 0.0475.
3. **Percentage:**
0.0475 * 100% = 4.75%
**ii. Packets more than 2030 grams:**
1. **Calculate the z-score:**
z = (2030 - 2025) / 15
z = 0.33
2. **Find the probability:**
The probability of a z-score greater than 0.33 is 1 - P(z < 0.33). Using a z-table, P(z < 0.33) is approximately 0.6293. So, the probability of a z-score greater than 0.33 is 1 - 0.6293 = 0.3707.
3. **Percentage:**
0.3707 * 100% = 37.07%
**iii. Adjusting the mean for 10% underweight:**
1. **Find the z-score for the 10th percentile:**
Using a z-table or calculator, the z-score corresponding to 10% (0.10) is approximately -1.28.
2. **Use the z-score formula to find the new mean (μ_new):**
x = μ_new + zσ
2000 = μ_new + (-1.28)(15)
2000 = μ_new - 19.2
μ_new = 2000 + 19.2
μ_new = 2019.2 grams
**iv. Adjusting the standard deviation for 10% less than 1995 grams:**
1. **Find the z-score for the 10th percentile:**
As before, the z-score for 10% is approximately -1.28.
2. **Use the z-score formula to find the new standard deviation (σ_new):**
x = μ + zσ_new
1995 = 2025 + (-1.28)σ_new
-30 = -1.28σ_new
σ_new = 30 / 1.28
σ_new ≈ 23.44 grams
**Summary of Answers:**
* i. Approximately 4.75% of packets are less than 2 kg.
* ii. Approximately 37.07% of packets are more than 2030 grams.
* iii. The manufacturer should adjust the mean to approximately 2019.2 grams.
* iv. The manufacturer should adjust the standard deviation to approximately 23.44 grams.
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