Thinning of the protective layer of ozone surrounding the earth may have catastr

Question

Thinning of the protective layer of ozone surrounding the earth may have catastrophic consequences. A team of University of California scientists estimated that increased solar radiation through the hole in the ozone layer over Antarctica altered processes to such an extent that primary production of phytoplankton was reduced 6 to 12%. Depletion of the ozone layer allows the most damaging ultraviolet radiation—UVB (280– 320 nm)—to reach the earth’s surface. An important consequence is the degree to which oceanic phytoplankton production is inhibited by exposure to UVB, both near the ocean surface (where the effect should be slight) and below the surface (where the effect could be considerable). To measure this relationship, the researchers sampled from the ocean column at various depths at 17 locations around Antarctica during the austral spring of 1990. To account for shifting of the ozone hole’s positioning, they constructed a measure of UVB exposure integrated over exposure time. The exposure measurements and the percentages of inhibition of normal phytoplankton production were extracted from their graph to produce Display 10.22. (Data from R. C. Smith et al., “Ozone Depletion: Ultraviolet Radiation and Phytoplankton Biology in Antarctic Waters,”Science 255 (1992): 952–57.) Does the effect of UVB exposure on the distribution of percentage inhibition differ at the surface and in the deep? How much difference is there?

Fit the regression model with response variable: Inhibit; and explanatory variables: UVB, Deep, Deep×UVB, where Deep is an indicator you create for the categorical variable Surface. Write separate regression models and fitted models for (i) Surface_Deep, and (ii) Surface_Surface. Explain which regression coefficient(s) should be tested in order to assess whether the two lines are parallel. Do the test and discuss. Are the intercept, and the coefficient of Surface_Deep significant? Discuss. Fit a model without an intercept for Inhibit using only UVB and Deep×UVB as predictors and discuss.

Inhibit UVB Surface 0 0 Deep 1 0 Deep 6 0.01 Deep 7 0.01 Surface 7 0.02 Surface 7 0.03 Surface 9 0.04 Surface 9.5 0.01 Deep 10 0 Deep 11 0.03 Surface 12.5 0.03 Surface 14 0.01 Deep 20 0.03 Deep 21 0.04 Surface 25 0.02 Deep 39 0.03 Deep 59 0.03 Deep

Explanation / Answer

Using MINITAB the two regression lines are

Regression Equation

Surface_Deep_1
1 Inhibit_1 = 1.18056 + 1226.39 UVB_1

Coefficients

Term Coef SE Coef T P
Constant 1.18 5.203 0.22690 0.826
UVB_1 1226.39 282.170 4.34627 0.002

Summary of Model

S = 10.7076 R-Sq = 70.25% R-Sq(adj) = 66.53%
PRESS = 1695.59 R-Sq(pred) = 45.00%

So the model for surface deep is Inhibit_1 = 1.18056 + 1226.39 UVB_1

(ii)

Surface_Surface_2
0 Inhibit_2 = 2.45833 + 286.458 UVB_2

Coefficients

Term Coef SE Coef T P
Constant 2.458 5.088 0.48314 0.649
UVB_2 286.458 168.278 1.70229 0.149

Summary of Model

S = 4.40655 R-Sq = 36.69% R-Sq(adj) = 24.03%
PRESS = 217.593 R-Sq(pred) = -41.89%

so model for surface_surface is  Inhibit_2 = 2.45833 + 286.458 UVB_2

(c)For testing whether the lines are parallel or not we need to check whether the slopes are equal or not. i.e for a y=ai + bi*xj ; i=1 and 2 are 2 different model then for testing parallel hypothesis is

H0:b1=b2

so we do significance test

here is the combined model output from minitab

General Regression Analysis: Inhibit versus UVB, Surface

Regression Equation

Surface
Deep Inhibit = 1.18056 + 1226.39 UVB

Surface Inhibit = 2.45833 + 286.458 UVB

Coefficients

Term Coef SE Coef T P
Constant 1.819 5.533 0.32884 0.748
UVB 756.424 204.920 3.69132 0.003
Surface
Deep -0.639 5.533 -0.11547 0.910
Surface*UVB
Deep 469.965 204.920 2.29341 0.039

Summary of Model

S = 8.83311 R-Sq = 70.86% R-Sq(adj) = 64.14%
PRESS = 1913.19 R-Sq(pred) = 45.04%

Analysis of Variance

Source DF Seq SS Adj SS Adj MS F P
Regression 3 2466.66 2466.66 822.22 10.5381 0.000868
UVB 1 750.00 1063.14 1063.14 13.6258 0.002715
Surface 1 1306.27 1.04 1.04 0.0133 0.909837
Surface*UVB 1 410.39 410.39 410.39 5.2597 0.039134
Error 13 1014.31 1014.31 78.02
Lack-of-Fit 4 72.64 72.64 18.16 0.1736 0.946440
Pure Error 9 941.67 941.67 104.63
Total 16 3480.97

As you can see p value for UVB is <.05, therefore slope are significant and they are not equal thus they are not parallel!

(d)intercept is not significant (p=.748 >.05) but coefficient is significant (<.05)

(e)the model without intercept is

The regression equation is
Inhibit = 363 UVB + 912 ubv*deep

MSE for this model is 68.31 and for the previous one is 78.02 so this can be regarded as a better fit!

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