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Question 1193540: (a) The regional manager of a bank wants to analyze the number of delayed repayments of
instalments of consumer loans in two of its branch banks (Branch −X and Branch −Y ).
The number of delayed payments of instalments in each branch bank follows normal
distribution. The manager feels that the number of delayed payments of instalments by
the consumers of the Branch −X is no way different from that of the Branch −Y . So, he
selected the loan accounts of 80 different consumers from the Branch −X and found that
the mean and variance of the number of delayed payments of instalments are 35 and 25,
respectively. Similarly, he selected the loan accounts of 100 different consumers from the Branch −Y and found that the mean and variance of the number of delayed payments of
instalments are 40 and 49, respectively. Test his intuition at a significance level of 0.01.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **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.**
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