The data in Figure 3 fit the regression Recall that the residuals a model of scatter about an invisible true deviations of the data points from the least- regression line reasonably well. The least- squares line squares line is y=91.27+1.493x.The residual = observed y-predicted y slope is particularly important. A slope is a =y-y rate of change. The true slope B says how There are n residuals, one for each data point. much higher average IQ is for children with Because o is the standard deviation of responses one more intensity unit in their crying about the true regression line, we estimate it by a measurement. Because b=1.493 sample standard deviation of the residuals. We call estimates the unknown B, we estimate that this sample standard deviation a standard error to on the average lQ is about 1.5 points higher emphasize that it is estimated from data. The for each added crying intensity residuals from a least-squares line always have mean zero. That simplifies their standard error.as Standard error about the least- We need the intercept a=91.27 to draw the line, but it has no statistical meaning in this squares line example. No child had fewer than 9 crying intensity, so we have no data near x=0 The standard error about the line The remaining parameter of the model is the standard deviation g, which describes the s=、 residual2 variability of the response y about the true regression line. The least-squares line estimates the true regression line. So the residuals estimate how much y varies about the true line Use s to estimate the unknown o in the17 33 • The data in Figure 3 fit the regression model of scatter about an invisible true regression line reasonably well. The least￾squares line is . The slope is particularly important. A slope is a rate of change. The true slope says how much higher average IQ is for children with one more intensity unit in their crying measurement. Because b = 1.493 estimates the unknown , we estimate that on the average IQ is about 1.5 points higher for each added crying intensity. yˆ = + 91.27 1.493 x β β 34 • We need the intercept a = 91.27 to draw the line, but it has no statistical meaning in this example. No child had fewer than 9 crying intensity, so we have no data near x = 0. • The remaining parameter of the model is the standard deviation , which describes the variability of the response y about the true regression line. The least-squares line estimates the true regression line. So the residuals estimate how much y varies about the true line. σ 18 35 • Recall that the residuals are the vertical deviations of the data points from the least￾squares line: residual observed predicted ˆ y y y y = − = − There are n residuals, one for each data point. Because is the standard deviation of responses about the true regression line, we estimate it by a sample standard deviation of the residuals. We call this sample standard deviation a standard error to emphasize that it is estimated from data. The residuals from a least-squares line always have mean zero. That simplifies their standard error. σ 36 Standard error about the least￾squares line • The standard error about the line is 2 2 1 residual 2 1 ()ˆ 2 s n y y n = − = − − ∑ ∑ Use s to estimate the unknown in the regression model. σ