Prediction a single new value of response

Although the derivation is beyond the scope of this presentation, it can be shown that the estimated variance of predicting a single new value of response at a given point in factor space, is equal to the purely experimental uncertainty variance, plus the variance of estimating the mean response at that point, 5, 0 that is. 

It is evident that Equation 11.71 can be used to plot the variance (or uncertainty) of predicting a single new value of response if Xq is made to vary across the domain of factor space. Such a plot of standard deviation of predicting a single new value of response as a function of pH is shown in Figure 11.7 for the experimental design of Equation 11.15, the data of Equation 11.16, and the second-order model of Equation 11.39. 

Figure 11.8 Confidence bands (95% level) for predicting a single new value of response (outer band), a single mean of four new values of response (middle band), and a single estimate of the true mean response (inner band).

Calculate the 99% confidence interval for predicting a single new value of response at pH = 7.0 for the data of Equation 11.16 and the second-order model of Equation 11.39. Calculate the 99% confidence interval for predicting the mean of seven new values of response for these conditions. Calculate the 99% confidence interval for predicting the true mean for these conditions. What confidence interval would be used if it were necessary to predict the true mean at several points in factor space ... 

The standard uncertainty in the single new value of response (the square root of the estimated variance of predicting the response, analogous to the standard uncertainty of a parameter estimate defined in Section 6.2) is... 

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