https://r2.erweima.ai/mathbot/image/172367675984463w19lvy.jpg

I'll begin by analyzing the image to extract the mathematical content.

### Analysis of the Image

The image appears to contain a word problem in Spanish about weekly income in the industry sector. Here's a translation and breakdown of the problem:

---

**Problem Statement:**

- The average weekly income of workers in the automotive industry is normally distributed.
- Mean income ($ \mu $) = $1,000
- Standard deviation ($ \sigma $) = $100

1. **Probability Calculation:**
   - **a.** The probability that a randomly selected worker earns between $900 and $1,100.
   - **b.** The probability that a randomly selected worker earns more than $1,100.

### Solutions

#### 1a. Probability of Earning Between $900 and $1,100

To find this probability, we will calculate the Z-scores for $900 and $1,100 and use the standard normal distribution table.

- **Z-score calculation:**
  \[
  Z = \frac{{X - \mu}}{{\sigma}}
  \]
  - For $900:
    \[
    Z_{900} = \frac{{900 - 1000}}{{100}} = -1
    \]
  - For $1,100:
    \[
    Z_{1100} = \frac{{1100 - 1000}}{{100}} = 1
    \]

- **Using Z-tables:**
  - Probability of $ Z \leq 1 $ ≈ 0.8413
  - Probability of $ Z \leq -1 $ ≈ 0.1587

- **Probability between $900 and $1,100:**
  \[
  P(900 \leq X \leq 1100) = P(Z \leq 1) - P(Z \leq -1) = 0.8413 - 0.1587 = 0.6826
  \]

#### 1b. Probability of Earning More Than $1,100

- **Using Z-tables:**
  \[
  P(X > 1100) = 1 - P(Z \leq 1) = 1 - 0.8413 = 0.1587
  \]

### Summary of Results

1. The probability that a randomly selected worker earns between $900 and $1,100 is **68.26%**.
2. The probability that a randomly selected worker earns more than $1,100 is **15.87%**.

If you have any questions or need further details, please let me know.

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Here are some related questions to explore further:

1. What are Z-scores, and how are they used in probability?
2. How does the normal distribution apply to real-world data?
3. How can we calculate probabilities for non-standard distributions?
4. What tools can be used for calculating statistical probabilities?
5. How do standard deviations affect the shape of a distribution?
6. What are the limitations of using normal distribution for all datasets?
7. How can we test if a dataset follows a normal distribution?
8. What are other common probability distributions?

**Tip:** Familiarize yourself with statistical tables and software tools for quicker probability calculations.

Learn how to calculate probabilities in a normal distribution problem involving weekly income in the automotive industry. Includes step-by-step solutions using Z-scores and standard normal distribution tables. Suitable for college-level students and professionals.

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