The regression model The mean response !, has a straight-line relationship with: The slope b and intercept a of the least- quare line are statistics. That is, we =a+Bx calculated them from the sample data These statistics would take somewhat different values if we repeated the study The slope B and intercept a are unknown with different infants To do formal inference, we think of a and b as estimates The standard deviation of y(call it o )is of unknown parameters the same for all values of x the value of o unknown Assumptions for regression The heart of this model is that there is an inference on the average" straight-line relationship between y and X. The true regression line We have n observations on an explanatory u,=a+Bx says that the mean variable x and a response variable yOur response u, moves along a straight line goal is to study or predict the behavior of y the explanatory variable x changes. We for given values of x cant observe the true regression line. The For any fixed value of x, the response y values of y that we do observe vary about varies according to a normal distribution their means according to a normal Repeated responses y are independent of distribution If we hold x fixed and take each other many observations on y, the normal pattern will eventually appear in a histogram13 25 The regression model • The slope b and intercept a of the least￾squares line are statistics. That is, we calculated them from the sample data. These statistics would take somewhat different values if we repeated the study with different infants. To do formal inference, we think of a and b as estimates of unknown parameters. 26 Assumptions for regression inference We have n observations on an explanatory variable x and a response variable y. Our goal is to study or predict the behavior of y for given values of x. • For any fixed value of x, the response y varies according to a normal distribution. Repeated responses y are independent of each other. 14 27 • The mean response has a straight-line relationship with x: The slope and intercept are unknown parameters. • The standard deviation of y (call it ) is the same for all values of x. The value of is unknown. y μ = + α β x μ y σ σ β α 28 The heart of this model is that there is an "on the average" straight-line relationship between y and x. The true regression line y μ = + α β x response moves along a straight line as the explanatory variable x changes. We can't observe the true regression line. The values of y that we do observe vary about their means according to a normal distribution. If we hold x fixed and take many observations on y, the normal pattern will eventually appear in a histogram. says that the mean μ y