Problem

Use the displayed table, representative factor \(0.80\), and \(\text{residual}~=~\text{observed}~-~\text{predicted}\) to evaluate an exponential fit. Calculate the ratios, model and percent change, predictions and residuals, then state whether the fit is exact or approximate.

Table of observed points (0, 1000), (1, 801), (2, 640), and (3, 512).

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ratios = 0.801,640/801≈0.7990,0.80; model = f(x)=1000·0.80ˣ; rate = about 20% decay/interval; predictions = 1000,800,640,512; residuals = 0,1,0,0; conclusion = good approximate exponential fit with a 1-unit deviation at x=1

ratios = 0.801,640/801≈0.7990,0.80; model = f(x)=1000·0.20ˣ; rate = 80% decay/interval; predictions = 1000,200,40,8; residuals = 0,601,600,504; conclusion = poor exponential fit using factor 0.20

ratios = 0.801,640/801≈0.7990,0.80; model = f(x)=1000·0.80ˣ; rate = 20% decay/interval; predictions = 1000,801,640,512; residuals = 0,0,0,0; conclusion = exact fit because observed values are used as predictions

ratios = 0.801,640/801≈0.7990,0.80; model = f(x)=1000·0.80ˣ; rate = 80% decay/interval; predictions = 1000,800,640,512; residuals = 0,1,0,0; conclusion = good approximate fit with a 1-unit deviation at x=1

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Use the displayed table, representative factor \(0.80\), and \(\text{residual}~=~\text{observed}~-~\text{predicted}\) to evaluate an exponential fit. Calculate the ratios, model and percent change, predictions and residuals, then state whether the fit is exact or approximate.

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