Explain P-values and Confidence Intervals in Simple Linear Regression

Discuss p-values, confidence intervals and simple linear regression. To make the relationship precise, use the slope in a regression of an outcome on a single predictor as your example. Define the model and the null hypothesis being tested, explain how a slope estimate and its uncertainty support inference, and interpret a corresponding confidence interval.

State the assumptions behind the inferential procedure you choose. Distinguish a confidence interval for a mean relationship from a prediction interval for a new outcome, and explain why neither a small p-value nor a fitted slope alone establishes that changing the predictor would cause the outcome to change. No dataset, estimated effect or numerical result is supplied.