9.3 Setting Up a Test for the Slope of a Regression Model

• The null hypothesis in testing the slope of a regression model is that the true slope is zero.


  True
• Match the assumption with its corresponding condition:

  Linearity ↔️ Scatterplot shows a linear pattern

  Independence ↔️ Residuals are independent

  Normality ↔️ Residuals are normally distributed

  Equal Variance ↔️ Homoscedasticity
• In the formula for the test statistic, SE(b) refers to the standard error of the slope
• Match the alternative hypothesis with its corresponding p-value formula:

  Ha: β > 0 ↔️ P(t > t-statistic)

  Ha: β < 0 ↔️ P(t < t-statistic)

  Ha: β ≠ 0 ↔️ 2 * P(|t| > |t-statistic|)
• The assumption of equal variance in residuals is also known as homoscedasticity
• The test statistic for the slope of a regression model is denoted by the variable t
• The t statistic measures how far the sample slope deviates from zero, relative to its variability.


  True
• Failing to reject the null hypothesis proves that it is true.

  False
• What decision is made if the p-value is less than or equal to the significance level (α)?

  Reject H0
• The alternative hypothesis in testing the slope of a regression model is that the true slope is not zero
• Steps to verify assumptions for a t-test of the slope of a regression model

  1️⃣ Examine the scatterplot for linearity

  2️⃣ Ensure data points are unrelated for independence

  3️⃣ Use a histogram or normal probability plot for normality

  4️⃣ Check residuals vs. predicted values for equal variance
• The degrees of freedom for the t-distribution in a test of the slope of a regression model are calculated as n - 2.


  True
• Checking assumptions for a t-test ensures the reliability and accuracy of the results.

  True
• The linearity condition requires that the scatterplot of the data shows a linear pattern.


  True
• What does the variable 'b' represent in the formula for the t-test statistic?

  Sample slope
• Match the alternative hypothesis with the corresponding p-value calculation:

  β > 0 (right-tailed) ↔️ Probability that t ≥ test statistic

  β < 0 (left-tailed) ↔️ Probability that t ≤ test statistic

  β ≠ 0 (two-tailed) ↔️ 2 × Probability that |t| ≥ |test statistic|
• What is the null hypothesis when testing the slope of a regression model?

  β = 0
• Steps in testing the slope of a regression model

  1️⃣ Define the null and alternative hypotheses

  2️⃣ Check assumptions and conditions

  3️⃣ Calculate the test statistic

  4️⃣ Determine the p-value

  5️⃣ Make a decision based on the p-value

  6️⃣ State the conclusion in context
• Checking assumptions such as linearity, independence, normality, and equal variance is crucial before testing the slope of a regression model.

  True
• The test statistic for the slope of a regression model is calculated using the formula: t = (b - β) / SE(b).

  True
• For a two-sided test, the p-value is twice the one-sided p-value
• What does it mean for residuals to be normally distributed in a t-test for the slope of a regression model?

  They follow a normal distribution
• What does it mean for residuals to be independent in a regression model?

  No correlation between errors
• In the t-test formula, the null hypothesis value (β) is usually assumed to be 0
• The significance level (α) is typically set to 0.05
• For a two-sided alternative hypothesis, the p-value is doubled to account for both tails of the distribution.


  True