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Thus, the true regression model Y 0 x 0 1x is a line of mean values; that is, the height of the regression line at any value of x is just the expected value of Y for that x. The slope, 1, can be interpreted as the change in the mean of Y for a unit change in x. Furthermore, the variability of Y at a particular value of x is determined by the error variance 2. This implies that there is a distribution of Y-values at each x and that the variance of this distribution is the same at each x. For example, suppose that the true regression model relating oxygen purity to hydrocarbon level is Y 0 x 75 15x, and suppose that the variance is 2 2. Figure 11-2 illustrates this situation. Notice that we have used a normal distribution to describe the random variation in . Since Y is the sum of a constant 0 1x (the mean) and a normally distributed random variable, Y is a normally distributed random variable. The variance 2 determines the

_message) As String) _message)

y (Oxygen purity)

0 + 1 (1.25)

2 3 4 1 B

_message)

0 + 1 (1.00)

x = 100

x = 1.25

15

x (Hydrocarbon level)

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variability in the observations Y on oxygen purity. Thus, when 2 is small, the observed values of Y will fall close to the line, and when 2 is large, the observed values of Y may deviate considerably from the line. Because 2 is constant, the variability in Y at any value of x is the same. The regression model describes the relationship between oxygen purity Y and hydrocarbon level x. Thus, for any value of hydrocarbon level, oxygen purity has a normal distribution with mean 75 15x and variance 2. For example, if x 1.25, Y has mean value Y x 75 15(1.25) 93.75 and variance 2. In most real-world problems, the values of the intercept and slope ( 0, 1) and the error variance 2 will not be known, and they must be estimated from sample data. Then this tted regression equation or model is typically used in prediction of future observations of Y, or for estimating the mean response at a particular level of x. To illustrate, a chemical engineer might be interested in estimating the mean purity of oxygen produced when the hydrocarbon level is x 1.25%. This chapter discusses such procedures and applications for the simple linear regression model. 12 will discuss multiple linear regression models that involve more than one regressor.

End Sub Public Overloads Sub Send( _ ByVal sendTo As String, _ ByVal subject As String, _ ByVal message As String) _sendTo = sendTo _subject = subject _message = message Send(_sendFrom, _sendTo, _subject, _message) End Sub Public Overloads Overrides Sub Send( _ ByVal sendTo As String, _ ByVal sendFrom As String, _ ByVal subject As String, _ ByVal message As String) _sendTo = sendTo _sendFrom = sendFrom _subject = subject _message = message MyBase.Send(_sendFrom, _sendTo, _subject, _message) End Sub Public Sub SendMessageTo(ByVal sendTo As String) _sendTo = sendTo Send(_sendFrom, _sendTo, _subject, _message) End Sub End Class

11-2

The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y. Suppose that the true relationship between Y and x is a straight line and that the observation Y at each level of x is a random variable. As noted previously, the expected value of Y for each value of x is E1Y 0 x2

The SendMessageTo method also was added to allow the same message to be sent to a different recipient.

0 1x

where the intercept 0 and the slope 1 are unknown regression coef cients. We assume that each observation, Y, can be described by the model Y

Both projects were given a version number of 1.0.0.0 and a strong name, and installed into the Global Assembly Cache. It s important to note that any assembly that has a strong name must reference assemblies that have a strong name. The accompanying sidebar, Getting Assemblies to Show in the .NET Tab, explains how to add entries to the Registry, which exposes the assemblies on the .NET tab when adding assembly references to a project. A new Web project is created that displays a feedback form to the user. When the user submits the feedback form, the message is sent to two recipients. The feedback page contains TextBoxes for the Name, Subject, and Message, as shown in Figure 15.17.

0 1x

Click the Home tab on the Ribbon. Click the Sort & Filter button. Click Custom Sort. The Sort dialog box appears.

(11-2)

where is a random error with mean zero and (unknown) variance 2. The random errors corresponding to different observations are also assumed to be uncorrelated random variables.

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