Problem Details:

We analyze the relationship between the selling price of a used car and its age using a linear regression model $\text{Price} = \beta_0 + \beta_1 \text{Age} + \epsilon$, based on the data provided in the table. Below are the solutions to each question.

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(a) Hypotheses for Linear Relationship

To test if there is a linear relationship between the selling price and age of a used car:

• Null hypothesis ($H_0$): $\beta_1 = 0$ (no linear relationship between age and selling price).
• Alternative hypothesis ($H_a$): $\beta_1 \neq 0$ (a linear relationship exists between age and selling price).

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(b) Calculate the Test Statistic

The test statistic for $\beta_1$ is computed as:
$t = \frac{\text{Coefficient of Age}}{\text{Standard Error of Age}}$
From the table: $\text{Coefficient of Age} = -1207.26, \quad \text{Standard Error of Age} = 124.96$
Substitute: $t = \frac{-1207.26}{124.96} \approx -9.662$

Answer: $t = -9.662$

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(c) Significance of Age at the 1% Level

At the 1% significance level, we compare the calculated $t$-statistic ($-9.662$) to the critical $t$-value from the $t$-distribution with $df = n - 2 = 18$.

Using a $t$-table or software:

• For a two-tailed test at $\alpha = 0.01$, $t_{\text{critical}} \approx \pm 2.878$.

Since $|t| = 9.662 > 2.878$, we reject $H_0$. The age of a used car is statistically significant in explaining its selling price.

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(d-1) Hypothesis Test for $\beta_1 = -1000$

To test whether $\beta_1$ differs from $-1000$, the hypotheses are:

• Null hypothesis ($H_0$): $\beta_1 = -1000$
• Alternative hypothesis ($H_a$): $\beta_1 \neq -1000$

The test statistic is: $t = \frac{\text{Coefficient of Age} - \text{Hypothesized Value}}{\text{Standard Error}}$
Substitute values: $t = \frac{-1207.26 - (-1000)}{124.96} = \frac{-1207.26 + 1000}{124.96} = \frac{-207.26}{124.96} \approx -1.659$

Answer: $t = -1.659$

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(d-2) Find the p-value

Using a $t$-distribution with $df = 18$, the two-tailed p-value for $t = -1.659$ can be found using statistical software or a table. Approximate values:

• $p \approx 0.05$.

Thus, $0.05 < p < 0.10$.

Answer: $0.05 < p < 0.10$

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(d-3) Conclusion

At the 1% significance level ($\alpha = 0.01$):

• The p-value ($\approx 0.05$) is greater than 0.01, so we fail to reject $H_0$.
• There is not enough evidence to conclude that $\beta_1$ differs from $-1000$.

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Summary Answers:

• (a): $H_0: \beta_1 = 0$, $H_a: \beta_1 \neq 0$.
• (b): $t = -9.662$.
• (c): Yes, age is significant at the 1% level.
• (d-1): $t = -1.659$.
• (d-2): $0.05 < p < 0.10$.
• (d-3): Fail to reject $H_0$; not enough evidence to conclude $\beta_1 \neq -1000$.

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Follow-Up Questions

1. Why is a two-tailed test used for this hypothesis?
2. How does the p-value relate to the significance level in decision-making?
3. What would happen if the significance level were 5% instead of 1%?
4. How does sample size impact the $t$-statistic and p-value?
5. Could multicollinearity impact the interpretation of the regression?

Tip: When interpreting p-values, always contextualize them with the significance level to make clear, data-driven conclusions.