Conduct a hypothesis test at the 1% significance level in order to determine if β1 differs from −2,000.

Note: Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.
 	Coefficients	Standard Error	t Stat	p-Value
Intercept	21,211.93	745.43	28.456	1.98E-19
Age	−1,206.24	120.93	 	8.01E-10


To determine whether \(\beta_1\) differs from \(-2000\) at the 1% significance level, we conduct a **hypothesis test** using the following steps:

### **1. Hypotheses**
- **Null Hypothesis (\(H_0\))**: \(\beta_1 = -2000\)
- **Alternative Hypothesis (\(H_a\))**: \(\beta_1 \neq -2000\)

### **2. Test Statistic**
The test statistic for the hypothesis test is calculated as:
\[
t = \frac{\hat{\beta}_1 - \text{Hypothesized Value}}{\text{Standard Error}}
\]

Given:
- \(\hat{\beta}_1 = -1206.24\)
- Hypothesized Value = \(-2000\)
- Standard Error = \(120.93\)

\[
t = \frac{-1206.24 - (-2000)}{120.93} = \frac{793.76}{120.93} \approx 6.562
\]

### **3. Critical Value**
At the \(1\%\) significance level (\(\alpha = 0.01\)) and given this is a two-tailed test, we compare the test statistic to the critical value from the \(t\)-distribution. 

- Degrees of freedom (\(df\)) is not specified. If a large sample size is assumed, the critical value approximates the \(z\)-distribution's value of \(\pm 2.576\).

### **4. Decision Rule**
- If \(|t| > 2.576\), reject \(H_0\).
- If \(|t| \leq 2.576\), fail to reject \(H_0\).

### **5. p-Value Approach**
The p-value associated with the calculated \(t\)-statistic (\(t = 6.562\)) is much smaller than \(0.01\) (already evident from the very small p-value for Age given in the problem, \(8.01 \times 10^{-10}\)).

### **Conclusion**
Since \(|t| = 6.562 > 2.576\) and the p-value (\(8.01 \times 10^{-10}\)) is less than \(0.01\), we **reject the null hypothesis**. There is sufficient evidence to conclude that \(\beta_1\) differs from \(-2000\).

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Do you want a deeper explanation of the calculations or the reasoning behind the hypothesis test? 

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### Related Questions:
1. How does the choice of significance level impact the hypothesis test?
2. What assumptions are made in performing this hypothesis test?
3. Can a one-tailed test be used in this scenario? Why or why not?
4. How would the result change if the significance level were 5% instead of 1%?
5. What does rejecting \(H_0\) imply about the relationship between Age and the response variable?

**Tip**: Always verify the assumptions of the hypothesis test, such as the normality of residuals and independence, for reliable results!

Conduct a hypothesis test at the 1% significance level in order to determine if β1 differs from −2,000. Note: Negative value should be indicated by a minus sign. Round your answer to 3 decimal places. Coefficients Standard Error t Stat p-Value Intercept 21,211.93 745.43 28.456 1.98E-19 Age −1,206.24 120.93 8.01E-10

Learn how to conduct a hypothesis test at the 1% significance level to determine if β1 differs from −2,000. This example includes key concepts like the t-test, statistical inference, and hypothesis testing, with a detailed step-by-step solution. Ideal for advanced high school students or college-level courses.

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