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Question 1196641: THE LINK FOR THE CONTINGENCY TABLE : https://imagizer.imageshack.com/img922/7664/ifwzWf.jpg
A department store is developing a new advertising campaign for their new location in a different​ region, and their marketing managers need to understand their target market better. A survey of adult shoppers found the probabilities that an adult would shop at their new store classified by age is shown in the contingency table. Are age and shopping at the department store​ independent? Explain.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! To determine whether age and shopping at the department store are independent, we need to test for statistical independence between the two variables.
### **Steps to Analyze Independence**
1. **Review the Contingency Table**
The table will display the joint probabilities (e.g., \( P(\text{Age Group} \cap \text{Shops at Store}) \)) and marginal probabilities (e.g., \( P(\text{Age Group}) \) and \( P(\text{Shops at Store}) \)).
2. **Independence Condition**
Age and shopping at the store are independent if:
\[
P(\text{Age Group} \cap \text{Shops at Store}) = P(\text{Age Group}) \cdot P(\text{Shops at Store})
\]
for all combinations of age groups and shopping preferences.
3. **Calculate Probabilities**
- Calculate the marginal probabilities (row and column totals).
- Compare the observed joint probabilities with the product of the corresponding marginal probabilities.
4. **Explain the Result**
- If all observed probabilities match the product of marginals, the variables are independent.
- If not, the variables are dependent.
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### Example (Simplified Table)
| Age Group | Shops (Yes) | Shops (No) | Total |
|----------------|-------------|------------|--------|
| 18-34 | 0.20 | 0.10 | 0.30 |
| 35-54 | 0.25 | 0.15 | 0.40 |
| 55+ | 0.10 | 0.20 | 0.30 |
| **Total** | 0.55 | 0.45 | 1.00 |
- **Marginal Probabilities**:
- \( P(\text{Shops Yes}) = 0.55 \)
- \( P(\text{Shops No}) = 0.45 \)
- \( P(\text{Age 18-34}) = 0.30 \), \( P(\text{Age 35-54}) = 0.40 \), \( P(\text{Age 55+}) = 0.30 \)
- **Test Independence** for \( \text{Age 18-34} \) and \( \text{Shops Yes} \):
\[
P(\text{Age 18-34} \cap \text{Shops Yes}) = 0.20
\]
\[
P(\text{Age 18-34}) \cdot P(\text{Shops Yes}) = 0.30 \cdot 0.55 = 0.165
\]
Since \( 0.20 \neq 0.165 \), the variables are **not independent**.
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### **Conclusion**
You would conduct this analysis for all age groups and shopping preferences. If any joint probability differs from the product of the marginals, age and shopping at the department store are **dependent**.
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