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To solve these two questions about critical values in t-tests, we need to understand the context of each problem and apply the correct statistical principles.

### Question 14:
**Problem:**
Given the following hypothesis in a t-test, what is the critical value for rejecting the null hypothesis at a 5% significance level in a multiple regression model with 2 independent variables and a sample size of 32? Round your answer to three decimal digits.

**Hypothesis:**
\[ H_0: \beta_1 = 0 \quad \text{vs} \quad H_1: \beta_1 \neq 0 \]

**Solution:**
1. **Determine the degrees of freedom (df):**
   - In a multiple regression with \(k\) independent variables and a sample size \(n\), the degrees of freedom is \(df = n - k - 1\).
   - Here, \(n = 32\) and \(k = 2\), so:
     \[
     df = 32 - 2 - 1 = 29
     \]

2. **Find the critical value:**
   - For a two-tailed test at the 5% significance level (\(\alpha = 0.05\)), we split \(\alpha\) into two tails, so each tail has \(\alpha/2 = 0.025\).
   - Using a t-table or statistical software for \(df = 29\) and \(\alpha/2 = 0.025\), we find the critical t-value.
   - The critical t-value for \(df = 29\) at \(\alpha/2 = 0.025\) is approximately 2.045.

**Answer:**
\[ \text{Critical value} \approx 2.045 \]

### Question 15:
**Problem:**
Give the critical value of a two-sided t-test with a large number of degrees of freedom (\(n - k - 1 \geq 1000\)) at the 5% significance level. Give two decimal digits of precision.

**Solution:**
1. **Large degrees of freedom:**
   - When the degrees of freedom are very large (greater than 1000), the t-distribution approaches the standard normal distribution (Z-distribution).

2. **Find the critical value:**
   - For a two-tailed test at the 5% significance level (\(\alpha = 0.05\)), we split \(\alpha\) into two tails, so each tail has \(\alpha/2 = 0.025\).
   - Using the Z-table for \(\alpha/2 = 0.025\), the critical Z-value is approximately 1.96.

**Answer:**
\[ \text{Critical value} \approx 1.96 \]

In summary:
- For Question 14, the critical value for 29 degrees of freedom at a 5% significance level is approximately 2.045.
- For Question 15, with a large number of degrees of freedom (greater than 1000), the critical value is approximately 1.96.

If you need further details or have any questions, feel free to ask!

Here are some additional questions you might be interested in:
1. How do you interpret the critical value in the context of hypothesis testing?
2. What is the difference between a t-test and a Z-test?
3. How does sample size affect the critical value in a t-test?
4. What are the assumptions underlying a t-test?
5. How do you calculate the degrees of freedom for a t-test in different scenarios?

**Tip:** When dealing with hypothesis tests, always check the assumptions of the test to ensure the results are valid.

Understand critical values in t-tests through detailed examples and calculations. Explore how degrees of freedom and significance levels affect critical values in hypothesis testing.

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