What are the differences between a t-interval for slopes and a t-test for a slope?

T-interval: Estimates the range of plausible values for the true slope. | T-test: Tests a hypothesis about the value of the slope (typically if it is zero).

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What are the differences between a t-interval for slopes and a t-test for a slope?

T-interval: Estimates the range of plausible values for the true slope. | T-test: Tests a hypothesis about the value of the slope (typically if it is zero).

What are the differences between the null and alternative hypotheses in a t-test for a slope?

Null Hypothesis (H0H_0): Assumes there is no linear relationship (slope = 0). | Alternative Hypothesis (HaH_a): Claims there is a linear relationship (slope ≠ 0, slope > 0, or slope < 0).

What are the differences between correlation (r) and the slope (b) of a regression line?

Correlation (r): Measures the strength and direction of a linear relationship, but is unitless and does not predict values. | Slope (b): Predicts the change in the response variable for a one-unit change in the explanatory variable and has units.

What are the differences between deterministic and predictive language when interpreting regression results?

Deterministic language: Implies a certain outcome (e.g., 'studying one more hour will increase the score by 5 points'). | Predictive language: Acknowledges variability and makes predictions (e.g., 'studying one more hour is predicted to increase the score by 5 points').

What are the differences between the standard deviation of the residuals (s) and the standard error of the slope (SEbSE_b)?

Standard deviation of residuals (s): Measures the typical distance of the observed values from the regression line. | Standard error of the slope (SEbSE_b): Measures the variability of the sample slopes around the true population slope.

What is the definition of explanatory variable?

The explanatory variable (independent variable) is plotted on the x-axis and explains the patterns seen in a scatterplot; consider it the 'cause'.

What is the definition of response variable?

The response variable (dependent variable) is plotted on the y-axis and responds to the explanatory variable; consider it the 'effect'.

What is the definition of inference?

Inference uses sample data to make predictions or test claims about a population parameter, moving from describing data to making informed decisions.

What is a t-interval for slopes?

A confidence interval used to estimate the true slope of the population regression line, providing a range of plausible values.

What is a t-test for a slope?

A hypothesis test used to determine if there is a significant linear relationship between two variables by testing if the slope is significantly different from zero.

What is the general form of the least-squares regression line equation?

y^=a+bx\hat{y} = a + bx, where y^\hat{y} is the predicted value, a is the y-intercept, and b is the slope.

How do you calculate the correlation coefficient (r) from R-squared?

r=±R2r = \pm \sqrt{R^2}. The sign of r matches the sign of the slope.

What is the formula for the test statistic (t) in a t-test for the slope?

t=b0SEbt = \frac{b - 0}{SE_b}, where b is the sample slope and SEbSE_b is the standard error of the slope.

What is the formula for the confidence interval for the slope (b)?

b±t[objectObject]b \pm t^[object Object], where b is the sample slope, t^ is the critical t-value, and SEbSE_b is the standard error of the slope.

How to calculate degrees of freedom (df) for t-tests and t-intervals for slopes?

df=n2df = n - 2, where n is the number of data points.