Question 1164810
```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 ****.