How to test hypotheses about regression coefficients using their sampling distributions.

📚 Quick Study Guide

• 🔢 Null Hypothesis ($H_0$): The regression coefficient is zero, meaning the predictor variable has no effect on the response variable. $H_0: \beta = 0$
• 🧪 Alternative Hypothesis ($H_1$): The regression coefficient is not zero, suggesting the predictor variable does have an effect. This can be two-tailed ($H_1: \beta \neq 0$) or one-tailed ($H_1: \beta > 0$ or $H_1: \beta < 0$).
• 📊 Test Statistic: Calculated as $t = \frac{\hat{\beta} - \beta_0}{SE(\hat{\beta})}$, where $\hat{\beta}$ is the estimated coefficient, $\beta_0$ is the value under the null hypothesis (usually 0), and $SE(\hat{\beta})$ is the standard error of the estimated coefficient.
• 📈 Sampling Distribution: Under the null hypothesis, the test statistic follows a t-distribution with $n-p$ degrees of freedom, where $n$ is the sample size and $p$ is the number of predictors (including the intercept).
• 🤔 P-value: 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 ($\alpha$), reject the null hypothesis. Common significance levels are 0.05 and 0.01.
• 📝 Confidence Intervals: A range of values within which the true regression coefficient is likely to fall. Calculated as $\hat{\beta} \pm t_{\alpha/2, n-p} \cdot SE(\hat{\beta})$.

Practice Quiz

1. Question 1: What is the null hypothesis when testing the significance of a regression coefficient?
   1. A) The regression coefficient is equal to 1.
   2. B) The regression coefficient is not equal to 0.
   3. C) The regression coefficient is equal to 0.
   4. D) The regression coefficient is greater than 0.
2. Question 2: What distribution is typically used for testing hypotheses about regression coefficients?
   1. A) Normal distribution
   2. B) Chi-squared distribution
   3. C) F-distribution
   4. D) t-distribution
3. Question 3: The test statistic for a regression coefficient is calculated as $t = 2.5$ with a p-value of 0.02. If the significance level is 0.05, what is the correct decision?
   1. A) Fail to reject the null hypothesis.
   2. B) Reject the null hypothesis.
   3. C) Accept the null hypothesis.
   4. D) Increase the significance level.
4. Question 4: What does the standard error of a regression coefficient estimate?
   1. A) The bias of the coefficient estimate.
   2. B) The variability of the coefficient estimate.
   3. C) The mean of the coefficient estimate.
   4. D) The median of the coefficient estimate.
5. Question 5: A 95% confidence interval for a regression coefficient is [0.1, 0.5]. What can you conclude about the null hypothesis that the coefficient is zero at a significance level of 0.05?
   1. A) Reject the null hypothesis.
   2. B) Fail to reject the null hypothesis.
   3. C) Accept the null hypothesis.
   4. D) The confidence interval is irrelevant to the hypothesis test.
6. Question 6: What happens to the degrees of freedom for the t-distribution as the sample size increases?
   1. A) They decrease.
   2. B) They increase.
   3. C) They stay the same.
   4. D) They become infinite.
7. Question 7: In a multiple regression model with 100 observations and 3 predictor variables (including the intercept), what are the degrees of freedom for the t-distribution used to test individual coefficients?
   1. A) 97
   2. B) 99
   3. C) 100
   4. D) 96