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.