Question 1193886
**1. Probability Sampling Technique**

**Stratified Sampling**

* **Steps:**
    1. **Divide the population into strata:** In this case, the production sites in the four counties can be considered strata. 
    2. **Determine the sample size for each stratum:** Allocate the sample size to each stratum proportionally to its size within the population. Since the production ratio is 8:5:3:4, allocate samples to each county in this proportion. 
    3. **Randomly select units within each stratum:** Use a random number generator or a table of random numbers to select the required number of units (e.g., jeans) from each county's production.

**Advantages of Stratified Sampling:**

* Ensures representation from each production site.
* Improves the accuracy of the sample by reducing sampling error.
* Allows for analysis of differences between production sites.

**2. Revenue Generated for Each Product Line**

* **Calculate the number of units produced for each product:**
    * Skinny Jeans: 5000 units * 50% = 2500 units
    * Straight Leg Jeans: 5000 units * 35% = 1750 units
    * Jeggings: 5000 units * 15% = 750 units

* **Calculate revenue for each product and style:**
    * **Skinny Jeans:**
        * Stone Washed: 2500 units * 60% * $210 = $315,000
        * Plain: 2500 units * 40% * $185 = $185,000
    * **Straight Leg Jeans:**
        * Stone Washed: 1750 units * 60% * $185 = $194,250
        * Plain: 1750 units * 40% * $90 = $63,000
    * **Jeggings:**
        * Stone Washed: 750 units * 60% * $170 = $76,500
        * Plain: 750 units * 40% * $110 = $33,000

* **Total Revenue:** $315,000 + $185,000 + $194,250 + $63,000 + $76,500 + $33,000 = $866,750

**Comment on Business Status:**

* Based on the revenue figures, Indiana Inc. appears to be generating significant revenue. 
* Skinny jeans and stone-washed jeans are the highest-revenue generators. 
* Further analysis would be needed to assess profitability, market share, and competitive landscape to fully evaluate the business status.

**3. Probability of Skinny Jeans given Stone Washed**

* **Let:**
    * A: Event that the product is Skinny Jeans
    * B: Event that the product is Stone Washed

* **Given:**
    * P(A) = Probability of Skinny Jeans = 50% = 0.5
    * P(B) = Probability of Stone Washed = 60% = 0.6
    * P(A and B) = Probability of Skinny Jeans and Stone Washed = 0.5 * 0.6 = 0.3

* **Use Bayes' Theorem:**
    * P(A | B) = P(B | A) * P(A) / P(B)
    * P(A | B) = (P(A and B) / P(A)) * P(A) / P(B)
    * P(A | B) = P(A and B) / P(B) 
    * P(A | B) = 0.3 / 0.6 = 0.5

* **Therefore, the probability that the product is Skinny Jeans given that it is Stone Washed is 0.5 or 50%.**

I hope this comprehensive analysis is helpful!