Question 1164810: In grading pears into small, medium, and large, Nature Farms packs the pears that weigh more than 3.6 ounces in packages marked "large" and the pears that weigh less than 2.4 ounces into packages marked "small"; the remainder are packed in packages marked "medium." If a day's packaging contained 3.92% large and 5.94% small pears, determine the mean and the standard deviation for the pears' weights. Assume that the distribution of the weights is normal
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! ```python
import scipy.stats as stats
# Probabilities
p_large = 0.0392
p_small = 0.0594
# Cumulative probabilities for Z
cp_large = 1 - p_large # P(X < 3.6)
cp_small = p_small # P(X < 2.4)
# Z-scores
z1 = stats.norm.ppf(cp_large) # For 3.6
z2 = stats.norm.ppf(cp_small) # For 2.4
# System of equations:
# 3.6 = mu + z1 * sigma
# 2.4 = mu + z2 * sigma
# Subtracting: 1.2 = (z1 - z2) * sigma
sigma = 1.2 / (z1 - z2)
mu = 3.6 - z1 * sigma
print(f"{z1=}")
print(f"{z2=}")
print(f"{sigma=}")
print(f"{mu=}")
```
```text
z1=1.7600460440337644
z2=-1.559830253819668
sigma=0.36145925098953136
mu=2.963815075216468
```
To find the mean () and standard deviation () of the pears' weights, we use the properties of the normal distribution and set up a system of linear equations based on the given probabilities.
### 1. Identify the given information
* **Large pears:**
* **Small pears:**
### 2. Find the corresponding -scores
We use the standard normal distribution table (or inverse cumulative distribution function) to find the -scores that correspond to these probabilities.
* For **Large pears**:
The -score such that is ****.
* For **Small pears**:
The -score such that is ****.
### 3. Set up the equations
Using the formula :
1.
2.
### 4. Solve the system of equations
Subtract the second equation from the first to eliminate :
Now, substitute back into the first equation to find :
### Final Answer:
* **Mean ():** ounces
* **Standard Deviation ():** ounces
The distribution of the pears' weights is ****.
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