Twenty-six observations from the article “Multiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes” (see Exercise 14.16 of Section 14.2 in the textbook) were used to fit a multiple regression model relating y = number of fish at intake to the independent variables x₁ = water temperature (°C), x₂ = number of pumps running, x₃ = sea state (taking values 0, 1, 2, or 3), and x₄ = speed (knots). Partial Minitab output follows.

  

       

a. Construct a 95% confidence interval for β₃, the coefficient of x₃  sea state. Interpret the resulting interval.

  

b. Construct a 90% confidence interval for the mean change in y associated with a 1° increase in temperature when number of pumps, sea state, and speed remain fixed. 

  

REF PRB:

  

When coastal power stations take in large quantities of cooling water, it is inevitable that a number of fish are drawn in with the water. Various methods  have been designed to screen out the fish. The article “Multiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes” (Journal of Applied Ecology [1983]: 33–42) examined intake fish catch at an English power plant and several other variables thought to affect fish intake:

   

a. Interpret the values of b₁ and b₄.

b. What proportion of observed variation in fish intake can be explained by the model relationship?  

c. Estimate the value of s.

d. Calculate adjusted R². How does it compare to R² itself?

The regression equation is Y = 92.0-2.18X1 - 19.2X2-9.38X3 + 2.32X4 Coef Stdev t-ratio 42.07 2.19 1.087 -2.00 9.215 -2.08 4.356 -2.15 0.7686 3.02 R-sq(adj) = 27.3% Predictor Constant X1 X2 X3 X4 s = 10.53 91.98 -2.179 -19.189 -9.378 2.3205 R-sq = 39.0%