Photosynthetic efficiency is the percentage of light energy that is converted into chemical energy by a plant. The chemical energy is used by a plant to maintain the plant's activities. A sample of grass was used to determine the efficiency of the grass species. Table 8 shows the results of the study at various temperatures, measured in degrees Celsius. The efficiency is measured as a percent.

(a) Draw a scatter diagram of the data. What type of relation appears to exist between temperature and efficiency?

(b) Find the quadratic regression equation \(\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}\).

(c) Draw a residual plot against the fitted values, \(x\), and \(x^{2}\). Also, draw a boxplot of the residuals. Are there any problems with the model?

(d) Interpret the coefficient of determination.

(e) Does the \(F\)-test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=0\) ? Is either coefficient not significantly different from zero?

(f) Construct and interpret \(95 \%\) confidence and prediction intervals for \(x=18\) degrees Celsius.

Approach We will follow the same procedures that we used in multiple regression. Enter the data into StatCrunch to obtain the quadratic regression. The steps for obtaining polynomial regression equation using Minitab, Excel, and StatCrunch are given in the Technology Step-by-Step.

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Table 8 Temperature, .x Efficiency, y Temperature, x Efficiency, y -1.5 33 15 91 0 46 17 89 2.5 55 20 77 5 80 22 72 7 87 25 54 10 93 27 46 12 95