The bad debt ratio for a financial institution is defined to be the dollar value of loans defaulted divided by the total dollar value of all loans made. Suppose that a random sample of 7 Ohio banks is selected and that the bad debt ratios (written as percentages) for these banks are 8%, 7%, 8%, 6%, 8%, 7%, and 5%.

 

(a-1) Banking officials claim that the mean bad debt ratio for all Midwestern banks is 3.5 percent and that the mean bad debt ratio for Ohio banks is higher. Set up the null and alternative hypotheses needed to attempt to provide evidence supporting the claim that the mean bad debt ratio for Ohio banks exceeds 3.5 percent. (Round your answers to 1 decimal place.)

 



 

(a-2) Discuss the meanings of a Type I error and a Type II error in this situation.

 



 

(b-1) Assuming that bad debt ratios for Ohio banks are approximately normally distributed, use critical values and the given sample information to test the hypotheses you set up in part a by setting α equal to .01. (Round your answers to 3 decimal places.)

 



 

(b-2) Interpret the p-value of .0001 for the test.

 



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Let's break this down step by step:

### Part (a-1): Setting Up Hypotheses

To set up the null and alternative hypotheses for this problem, we will define the following:

- \( H_0 \): The null hypothesis is that the mean bad debt ratio for Ohio banks is 3.5%.
- \( H_1 \): The alternative hypothesis is that the mean bad debt ratio for Ohio banks is higher than 3.5%.

Mathematically, this can be expressed as:

- \( H_0: \mu = 3.5 \)
- \( H_1: \mu > 3.5 \)

### Part (a-2): Type I and Type II Errors

- **Type I Error**: This error occurs when we reject the null hypothesis when it is actually true. In this context, it means we would conclude that the mean bad debt ratio for Ohio banks is higher than 3.5% when it is actually 3.5%.
- **Type II Error**: This error occurs when we fail to reject the null hypothesis when it is actually false. In this context, it means we would conclude that the mean bad debt ratio for Ohio banks is not higher than 3.5% when it actually is higher.

### Part (b-1): Hypothesis Testing Using Critical Values

To perform the hypothesis test, we will follow these steps:

1. **Calculate the sample mean (\(\bar{x}\)) and the sample standard deviation (s)**:

   \[
   \bar{x} = \frac{8 + 7 + 8 + 6 + 8 + 7 + 5}{7} = \frac{49}{7} = 7
   \]

   \[
   s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(8-7)^2 + (7-7)^2 + (8-7)^2 + (6-7)^2 + (8-7)^2 + (7-7)^2 + (5-7)^2}{6}} = \sqrt{\frac{1 + 0 + 1 + 1 + 1 + 0 + 4}{6}} = \sqrt{\frac{8}{6}} = 1.155
   \]

2. **State the test statistic formula**:

   \[
   t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} = \frac{7 - 3.5}{1.155/\sqrt{7}} = \frac{3.5}{0.4366} = 8.015
   \]

3. **Determine the critical value for \( t \) at \( \alpha = 0.01 \) and \( df = n-1 = 6 \)**:
   
   Using a t-distribution table or calculator, the critical value at \( \alpha = 0.01 \) for a one-tailed test with 6 degrees of freedom is approximately \( t_{critical} = 3.143 \).

4. **Compare the test statistic to the critical value**:

   \[
   t_{calculated} = 8.015
   \]

   Since \( 8.015 > 3.143 \), we reject the null hypothesis.

### Part (b-2): Interpretation of the p-value

- A p-value of 0.0001 indicates that there is a very low probability that the observed data would occur if the null hypothesis were true. In this context, it means there is strong evidence against the null hypothesis, suggesting that the mean bad debt ratio for Ohio banks is indeed higher than 3.5%.

Would you like any further details or have any questions?

Here are some additional questions you might consider:

1. How do you interpret the confidence interval for the mean bad debt ratio?
2. What are the implications of a high bad debt ratio for banks?
3. How does the sample size affect the reliability of hypothesis testing?
4. What other statistical tests could be used to analyze this data?
5. How can banks use this statistical analysis to improve their financial stability?

**Tip:** Always check the assumptions of your statistical test, such as normality and independence of observations, to ensure the validity of your results.

Learn how to conduct hypothesis testing for the mean bad debt ratio of Ohio banks using sample data. This problem involves setting up hypotheses, calculating test statistics, and interpreting p-values. Suitable for college-level students and statisticians.

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