igure 4 The regression model. The line is the true regression line, which shows how the mean In practice, we observe y for many different response A, changes as the explanatory variable x values of x, so that we see an overall linear changes. For any fixed value of x, the observed pattern formed by points scattered about the response y varies according to a normal distribution true line. The standard deviation g aving mean p determines whether the points fall close to the true regression line(small a )or are widely scattered(large o Figure 4 shows the regression model in Inference about the model picture form. The line in the figure is the true regression line. The mean of the ponse y moves along this line as the The first step in inference is to estimate explanatory variable x takes different the unknown parameters a, B, and o values. The normal curves show how y will When the regression model describes our vary when x is held fixed at different data and we calculate the least-squares values. all of the curves have the same o so the variability of y is the same for all line y=a+bx, the slope b of the least. squares line is an unbiased estimator of values of x. you should check the assumptions for inference when you do the true slope B, and the intercept a of the least-squares line is an unbiased inference about regression estimator of the true intercept a15 29 In practice, we observe y for many different values of x, so that we see an overall linear pattern formed by points scattered about the true line. The standard deviation determines whether the points fall close to the true regression line (small ) or are widely scattered (large ). σ σ σ 30 • Figure 4 shows the regression model in picture form. The line in the figure is the true regression line. The mean of the response y moves along this line as the explanatory variable x takes different values. The normal curves show how y will vary when x is held fixed at different values. All of the curves have the same , so the variability of y is the same for all values of x. You should check the assumptions for inference when you do inference about regression. σ 16 31 Figure 4 The regression model. The line is the true regression line, which shows how the mean response changes as the explanatory variable x changes. For any fixed value of x, the observed response y varies according to a normal distribution having mean . μ y μ y 32 Inference about the Model • The first step in inference is to estimate the unknown parameters , , and . When the regression model describes our data and we calculate the least-squares line , the slope b of the least￾squares line is an unbiased estimator of the true slope , and the intercept a of the least-squares line is an unbiased estimator of the true intercept . α β σ y ˆ = + a bx β α