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What is the definition of t-distribution in the context of regression slope estimation?
When the assumptions for linear regression are met, and the null hypothesis is true, the distribution of the slope estimate follows a t-distribution with *n-2* degrees of freedom.
What is the definition of a t-score?
A t-score helps us understand how far our sample slope is from the null slope. It measures the evidence against the null hypothesis.
What is the definition of a p-value?
The p-value is the probability of observing a sample slope as extreme as (or more extreme than) the one we got, assuming the null hypothesis is true.
Define null hypothesis.
A null hypothesis is a statement that there is no effect or no relationship between variables. It is what we try to disprove.
Define alternative hypothesis.
An alternative hypothesis is a statement that there is an effect or a relationship between variables that the researcher is trying to find evidence for.
Explain the concept of degrees of freedom in the context of hypothesis testing for regression slope.
Degrees of freedom represent the number of independent pieces of information available to estimate parameters. For regression slope, it's *n-2* because we estimate both the slope and the intercept.
Explain the concept of significance level (alpha) in hypothesis testing.
The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true (Type I error). It is a threshold used to determine if the p-value is small enough to reject the null hypothesis.
Explain the relationship between the p-value and the decision to reject or fail to reject the null hypothesis.
If the p-value is less than the significance level (alpha), we reject the null hypothesis. If the p-value is greater than or equal to alpha, we fail to reject the null hypothesis.
Explain the meaning of 'rejecting the null hypothesis' in the context of regression slope.
Rejecting the null hypothesis means there is significant evidence to suggest that the true slope of the regression line is not zero, indicating a linear relationship between the variables.
Explain the meaning of 'failing to reject the null hypothesis' in the context of regression slope.
Failing to reject the null hypothesis means there is not enough evidence to suggest that the true slope of the regression line is different from zero, indicating no significant linear relationship between the variables.
What is the formula for calculating the t-score for the slope of a regression model?
$t = \frac{b - \beta}{SE_{b}}$
Where:
*b* is the sample slope.
*ฮฒ* is the hypothesized population slope (usually 0).
*SE<sub>b</sub>* is the standard error of the sample slope.
How do you calculate the degrees of freedom for a t-test of the slope?
Degrees of freedom = *n - 2*, where *n* is the sample size.