Question 1193378: 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 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 CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Set up Hypotheses**
* **Null Hypothesis (H0):** The mean number of delayed repayments of installments is the same for both branches.
* μ1 = μ2
* Where μ1 is the mean for Branch - X and μ2 is the mean for Branch - Y.
* **Alternative Hypothesis (H1):** The mean number of delayed repayments of installments is different for the two branches.
* μ1 ≠ μ2
**2. Determine the Test Statistic**
Since we are comparing the means of two independent samples with known variances, we can use the **Z-test for two independent samples**.
**3. Calculate the Test Statistic**
* **Sample Data:**
* Branch - X:
* Sample size (n1) = 80
* Sample mean (x̄1) = 35
* Sample variance (s1^2) = 25
* Branch - Y:
* Sample size (n2) = 100
* Sample mean (x̄2) = 40
* Sample variance (s2^2) = 49
* **Calculate the Z-statistic:**
Z = (x̄1 - x̄2) / √[(s1^2 / n1) + (s2^2 / n2)]
Z = (35 - 40) / √[(25 / 80) + (49 / 100)]
Z = -5 / √(0.3125 + 0.49)
Z = -5 / √0.8025
Z = -5 / 0.8958
Z ≈ -5.58
**4. Determine the Critical Value**
* Significance level (α) = 0.01
* This is a two-tailed test (since H1 is μ1 ≠ μ2)
* Find the critical Z-values from the standard normal distribution table:
* Zα/2 = Z0.005 = ±2.576
**5. Decision Rule**
* Reject H0 if the calculated Z-statistic (|Z|) is greater than the critical Z-value (2.576).
* Otherwise, fail to reject H0.
**6. Conclusion**
* Calculated Z-statistic (|-5.58|) = 5.58 is greater than the critical Z-value (2.576).
* **Therefore, we reject the null hypothesis (H0).**
**Interpretation:**
* There is sufficient evidence at the 0.01 significance level to conclude that the mean number of delayed repayments of installments differs between Branch - X and Branch - Y.
**Note:**
* This analysis assumes that the populations of delayed repayments for both branches are normally distributed.
* If the variances of the two populations are significantly different, a more appropriate test would be the Welch's t-test.
**Disclaimer:** This analysis provides a general framework. For real-world applications, consult with a qualified statistician or use statistical software for accurate calculations and interpretations.
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