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what is the critical value, what is the test statistic, what are the null and alternative hypothesis and what is the conclusion

Let's break down and solve the problem step by step:

### 1. **Linear Regression Equation**
The linear regression equation given is:

\[
f = 38.03 + 0.07 \cdot t
\]

Where:
- \( f \) is the predicted tuition fees.
- \( t \) is the time in years.
- \( \hat{a}_0 = 38.03 \) is the intercept.
- \( \hat{a}_1 = 0.07 \) is the slope coefficient, which represents the rate of change of tuition fees per year.

### 2. **Given Information**
We are also given:
- \( s_{\hat{a}_0} = 14.2 \) (standard error of the intercept),
- \( s_{\hat{a}_1} = 0.05 \) (standard error of the slope),
- \( r^2 = 0.41 \) (coefficient of determination).

### 3. **Hypotheses for the Slope**
The task is to verify if there is a correlation between time and tuition fees at the 1% significance level using a hypothesis test.

- **Null Hypothesis (\( H_0 \))**: The slope is zero, meaning there is no relationship between time and tuition fees.
  \[
  H_0: \hat{a}_1 = 0
  \]
  
- **Alternative Hypothesis (\( H_1 \))**: The slope is not zero, meaning there is a relationship between time and tuition fees.
  \[
  H_1: \hat{a}_1 \neq 0
  \]

### 4. **Test Statistic**
The test statistic for the slope (\( \hat{a}_1 \)) is calculated using the formula:

\[
t = \frac{\hat{a}_1 - 0}{s_{\hat{a}_1}} = \frac{0.07}{0.05}
\]

Let me calculate that:

\[
t = \frac{0.07}{0.05} = 1.4
\]

### 5. **Degrees of Freedom**
The degrees of freedom (\( df \)) for this test are:

\[
df = n - 2 = 55 - 2 = 53
\]

### 6. **Critical Value**
Since we are conducting a two-tailed t-test at a 1% significance level (\( \alpha = 0.01 \)), we need the critical t-value for \( df = 53 \) and \( \alpha = 0.01 \).

I will use a t-table to find the critical value. For \( df = 53 \), the critical t-value at the 1% significance level (two-tailed) is approximately:

\[
t_{critical} \approx 2.67
\]

### 7. **Conclusion**
- **Test Statistic**: The calculated test statistic is \( t = 1.4 \).
- **Critical Value**: The critical t-value is \( t_{critical} \approx 2.67 \).
- Since \( t = 1.4 \) is less than the critical value of \( 2.67 \), we fail to reject the null hypothesis at the 1% significance level.

### **Conclusion**:
There is not enough evidence to suggest a significant relationship between time and tuition fees at the 1% significance level.

---

Do you need any further details or explanations?

Here are 5 follow-up questions related to this:

1. What would the conclusion be if the significance level was 5% instead of 1%?
2. How is the \( r^2 \) value interpreted in this context?
3. How would we test for the significance of the intercept?
4. Can we predict tuition fees for years outside the 20-year range using this model? Why or why not?
5. How would the test results change if the sample size increased?

**Tip**: When performing hypothesis testing, always check whether the significance level and the test statistic align with the decision to reject or not reject the null hypothesis.

What is the critical value, what is the test statistic, what are the null and alternative hypothesis, and what is the conclusion?

Learn how to perform hypothesis testing on a linear regression model. This problem walks through calculating the test statistic and critical value, setting up null and alternative hypotheses, and determining the conclusion at a 1% significance level. Ideal for university-level students.

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