Question 1193540
**1. Set up Hypotheses**

* **Null Hypothesis (H0):** The mean number of delayed payments of installments is the same for both branches. 
    * μ₁ = μ₂ 
* **Alternative Hypothesis (H1):** The mean number of delayed payments of installments is different for the two branches.
    * μ₁ ≠ μ₂

**2. Choose the Test Statistic**

* Since we are comparing the means of two independent samples with known (or assumed) population variances, we will use the **Z-test for the difference between two means**.

**3. Calculate the Test Statistic**

* **Given:**
    * Sample 1 (Branch X): n₁ = 80, x̄₁ = 35, σ₁² = 25 
    * Sample 2 (Branch Y): n₂ = 100, x̄₂ = 40, σ₂² = 49
* **Calculate the pooled variance (not needed in this case since population variances are known):**
    * Pooled variance (s_p²) = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / (n₁ + n₂ - 2) 
* **Calculate the standard error of the difference between means:**
    * SE = √[(σ₁²/n₁) + (σ₂²/n₂)] 
        * SE = √[(25/80) + (49/100)] 
        * SE = √(0.3125 + 0.49) 
        * SE = √0.8025 
        * SE ≈ 0.8958

* **Calculate the Z-score:**
    * Z = (x̄₁ - x̄₂) / SE 
        * Z = (35 - 40) / 0.8958 
        * Z = -5 / 0.8958 
        * Z ≈ -5.58

**4. Determine Critical Values**

* **Significance Level:** α = 0.01
* **Two-tailed test:** We need to find the critical values for both tails of the standard normal distribution.
* **Using a standard normal distribution table or statistical software:**
    * Critical values: Z_critical ≈ ±2.576

**5. Decision Rule**

* If the calculated Z-score (|Z|) is greater than the critical value (Z_critical), reject the null hypothesis.
* If the calculated Z-score (|Z|) is less than or equal to the critical value (Z_critical), fail to reject the null hypothesis.

**6. Make a Decision**

* Our calculated Z-score (|-5.58|) is greater than the critical value (2.576).

* **Conclusion:** We reject the null hypothesis.

**Interpretation**

The evidence suggests that the mean number of delayed payments of installments is significantly different between Branch X and Branch Y at the 0.01 significance level. 

**Therefore, the manager's intuition that the number of delayed payments is the same for both branches is not supported by the data.**