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Explain the concept of statistical significance in the context of regression slopes.
Statistical significance means that the observed relationship between the variables in the sample is unlikely to have occurred by random chance alone, suggesting a real relationship exists in the population.
Explain why we test if the slope (β) is different from zero.
Testing if β ≠ 0 determines if there's a meaningful linear relationship between the variables. If β = 0, there's no linear association.
Explain the importance of checking conditions before conducting a t-test for slopes.
Checking conditions ensures that the assumptions underlying the t-test are met, which validates the reliability and accuracy of the test results and conclusions.
Explain the implication of a non-linear pattern in the residual plot.
A non-linear pattern in the residual plot suggests that a linear model is not appropriate for the data, and a different type of model might be needed.
Explain the implication of unequal variance in the residual plot.
Unequal variance (fanning) in the residual plot violates the assumption of homoscedasticity, which can lead to inaccurate standard errors and unreliable test results.
What is the null hypothesis (H₀) in a t-test for slopes?
The null hypothesis assumes there is no relationship (or a specific relationship) between the variables. It's in the form H₀: β = β₀.
What is the alternative hypothesis (Hₐ) in a t-test for slopes?
The alternative hypothesis is what we're trying to find evidence for – that there *is* a relationship. It takes the form of Hₐ: β ≠ β₀, Hₐ: β < β₀, or Hₐ: β > β₀.
Define the t-statistic in the context of regression slope testing.
A measure of how far the sample slope is from the hypothesized slope (usually zero), in terms of standard errors. A large t-statistic suggests a significant relationship.
What does 'β' represent in the context of hypothesis testing for regression slopes?
β represents the population slope, the true change in the dependent variable for each unit increase in the independent variable.
What is a residual plot used for in regression analysis?
A residual plot is a graph that displays the residuals on the y-axis and the independent variable on the x-axis. It is used to assess the linearity and equal variance conditions of a regression model.
What is the general form of the null hypothesis for the slope?
H₀: β = β₀, where β₀ is the hypothesized slope value.
Give the forms of the alternative hypothesis for the slope.
Hₐ: β ≠ β₀ (two-tailed), Hₐ: β < β₀ (left-tailed), or Hₐ: β > β₀ (right-tailed)