CONFIDENCE INTERVALS Con dence Intervals on the Slope and Intercept in .NET

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CONFIDENCE INTERVALS Con dence Intervals on the Slope and Intercept
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In addition to point estimates of the slope and intercept, it is possible to obtain con dence interval estimates of these parameters. The width of these con dence intervals is a measure of
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CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION
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the overall quality of the regression line. If the error terms, i, in the regression model are normally and independently distributed, 1 1 1 0
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are both distributed as t random variables with n 2 degrees of freedom. This leads to the following de nition of 100(1 )% con dence intervals on the slope and intercept. De nition Under the assumption that the observations are normally and independently distributed, a 100(1 )% con dence interval on the slope 1 in simple linear regression is
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EXAMPLE 11-4
We will nd a 95% con dence interval on the slope of the regression line using the data in Example 11-1. Recall that 1 14.947, Sxx 0.68088, and 2 1.18 (see Table 11-2). Then, from Equation 10-31 we nd
t0.025,18
or 14.947 This simpli es to 2.101
1.18 A 0.68088
B Sx x
t0.025,18
B Sx x
1.18 A 0.68088
11-6.2
Con dence Interval on the Mean Response
A con dence interval may be constructed on the mean response at a speci ed value of x, say, x0. This is a con dence interval about E(Y x0) Y x0 and is often called a con dence interval about the regression line. Since E(Y x0) Y x0 0 1x0, we may obtain a point estimate of the mean of Y at x x0 ( Y x0) from the tted model as
Y 0 x0
x 1 0
11-6 CONFIDENCE INTERVALS
Now Y 0 x0 is an unbiased point estimator of 0 and 1. The variance of Y 0 x0 is V 1 Y 0 x0 2
Y x0,
since 0 and 1 are unbiased estimators of 1x0 x2 2 Sx x
1 cn
This last result follows from the fact that cov 1Y, 1 2 0 (Refer to Exercise 11-71). Also, Y 0 x0 is normally distributed, because 1 and 0 are normally distributed, and if we 2 use as an estimate of 2, it is easy to show that B
Y 0 x0 2 c
Y 0 x0
Sx x
x 22
has a t distribution with n terval de nition. De nition A 100(1 x x0, say
Y 0 x0
2 degrees of freedom. This leads to the following con dence in-
)% con dence interval about the mean response at the value of Y 0 x0, is given by B
2 c
2, n 2
Y 0 x0
Sx x
x 22
2, n 2
Y 0 x0
2 c
Sx x
x 22
(11-31)
where Y 0 x0
x is computed from the tted regression model. 1 0
Note that the width of the con dence interval for Y 0 x0 is a function of the value speci ed for x0. The interval width is a minimum for x0 x and widens as 0 x0 x 0 increases. EXAMPLE 11-5 We will construct a 95% con dence interval about the mean response for the data in Example 11-1. The tted model is Y 0 x0 74.283 14.947x0, and the 95% con dence interval on Y 0 x0 is found from Equation 11-31 as
Y 0 x0
1.18 c
1 20
1.19602 2 d 0.68088 1.00%. Then
Suppose that we are interested in predicting mean oxygen purity when x0
Y 0 x1.00
74.283 B
and the 95% con dence interval is e 89.23 2.101
1.18 c
1 20
11.00 1.19602 2 df 0.68088
CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION
99 Oxygen purity y (%)
Figure 11-7 Scatter diagram of oxygen purity data from Example 11-1 with tted regression line and 95 percent con dence limits on Y 0 x0 .
87 0.87 1.07 1.27 Hydrocarbon level (%) x 1.47 1.67
or 89.23 Therefore, the 95% con dence interval on 88.48 0.75 is 89.98
Y 0 1.00 Y 0 1.00
Minitab will also perform these calculations. Refer to Table 11-2. The predicted value of y at x 1.00 is shown along with the 95% CI on the mean of y at this level of x. By repeating these calculations for several different values for x0 we can obtain con dence limits for each corresponding value of Y 0 x0. Figure 11-7 displays the scatter diagram with the tted model and the corresponding 95% con dence limits plotted as the upper and lower lines. The 95% con dence level applies only to the interval obtained at one value of x and not to the entire set of x-levels. Notice that the width of the con dence interval on Y 0 x0 increases as 0 x0 x 0 increases.
11-7
PREDICTION OF NEW OBSERVATIONS
An important application of a regression model is predicting new or future observations Y corresponding to a speci ed level of the regressor variable x. If x0 is the value of the regressor variable of interest,
Y0
x 1 0
(11-32)
is the point estimator of the new or future value of the response Y0. Now consider obtaining an interval estimate for this future observation Y0. This new observation is independent of the observations used to develop the regression model. Therefore, the con dence interval for Y 0 x0 in Equation 11-31 is inappropriate, since it is based only on the data used to t the regression model. The con dence interval about Y 0 x0 refers to the true mean response at x x0 (that is, a population parameter), not to future observations.