Common Mistakes When Performing Hypothesis Tests for β₁ Regression

📚 Quick Study Guide

• 📊 Null Hypothesis: The null hypothesis for $\beta_1$ is typically that there is no linear relationship between the independent and dependent variables, i.e., $H_0: \beta_1 = 0$.
• 🧪 Test Statistic: The test statistic is calculated as $t = \frac{\hat{\beta_1} - 0}{SE(\hat{\beta_1})}$, where $\hat{\beta_1}$ is the estimated coefficient and $SE(\hat{\beta_1})$ is its standard error.
• 📈 Degrees of Freedom: The degrees of freedom for the t-test are $n - 2$, where $n$ is the number of observations.
• 🤔 P-value: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
• ✅ Decision Rule: If the p-value is less than the significance level $\alpha$ (e.g., 0.05), we reject the null hypothesis.
• 💡 Confidence Interval: A confidence interval for $\beta_1$ is given by $\hat{\beta_1} \pm t_{\alpha/2, n-2} \cdot SE(\hat{\beta_1})$.
• 📉 Assumptions: Linear regression relies on assumptions such as linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of errors.

Practice Quiz

1. Which of the following is a common null hypothesis when testing for the significance of $\beta_1$ in linear regression?

   1. A) $H_0: \beta_1 > 0$
   2. B) $H_0: \beta_1 \neq 0$
   3. C) $H_0: \beta_1 = 0$
   4. D) $H_0: \beta_1 < 0$



2. What is the correct formula for the t-statistic used to test the hypothesis about $\beta_1$?

   1. A) $t = \frac{SE(\hat{\beta_1})}{\hat{\beta_1} - 0}$
   2. B) $t = \frac{\hat{\beta_1} - 0}{SE(\hat{\beta_1})}$
   3. C) $t = \hat{\beta_1} - 0 \cdot SE(\hat{\beta_1})$
   4. D) $t = \frac{\hat{\beta_1}}{SE(\hat{\beta_1})}$



3. In a simple linear regression with 30 observations, what are the degrees of freedom for the t-test of $\beta_1$?

   1. A) 30
   2. B) 29
   3. C) 28
   4. D) 31



4. What does a small p-value (e.g., p < 0.05) indicate when testing the hypothesis about $\beta_1$?

   1. A) Evidence in favor of the null hypothesis.
   2. B) Evidence against the null hypothesis.
   3. C) The null hypothesis is true.
   4. D) The alternative hypothesis is false.



5. If the 95% confidence interval for $\beta_1$ is (-0.5, 0.8), what can you conclude about the significance of $\beta_1$ at the 5% significance level?

   1. A) $\beta_1$ is statistically significant.
   2. B) $\beta_1$ is not statistically significant.
   3. C) We cannot determine the significance of $\beta_1$ from the confidence interval.
   4. D) $\beta_1$ is equal to 0.



6. Which of the following assumptions is NOT required for valid hypothesis testing in linear regression?

   1. A) Linearity
   2. B) Independence of errors
   3. C) Homoscedasticity
   4. D) Multicollinearity



7. What is the consequence of ignoring heteroscedasticity when performing a hypothesis test for $\beta_1$?

   1. A) It does not affect the validity of the test.
   2. B) The test becomes more powerful.
   3. C) The standard errors are biased, leading to incorrect p-values and confidence intervals.
   4. D) The estimated $\beta_1$ is biased.