Question

The invasive diatom species Didymosphenia geminata has the potential to inflict substantial ecological and economic damage in rivers. An article described an investigation of colonization behavior. One aspect of particular interest was whether

y = colony

density was related to

x = rock

surface area. The article contained a scatterplot and summary of a regression analysis. Here is representative data.

x	50	71	55	50	33	58	79	26
y	159	1936	55	29	9	12	42	14

x	69	44	37	70	20	45	49
y	276	45	178	20	50	192	32

(a) Fit the simple linear regression model to this data. (Round your numerical values to three decimal places.)

y =

Predict colony density when surface area = 70 and calculate the corresponding residual. (Round your answers to the nearest whole number.)

colony density
corresponding residual

Predict colony density when surface area = 71 and calculate the corresponding residual. (Round your answers to the nearest whole number.)

colony density
corresponding residual

How do the residuals compare?

The residuals for both points are positive. The residual for the first point is negative, while the residual for the second point is positive.     The residual for the first point is positive, while the residual for the second point is negative. The residuals for both points are negative.

(b) Calculate the coefficient of determination. (Round your answer to three decimal places.)


Interpret the coefficient of determination.

The coefficient of determination is the proportion of the total variation in rock surface area that can be explained by a linear regression model with colony density as the predictor. The coefficient of determination is the increase in rock surface area due to an increase in one unit of rock surface area.     The coefficient of determination is the increase in colony density due to an increase in one unit of colony density. The coefficient of determination is the probability that the regression model fits the data. The coefficient of determination is the proportion of the total variation in colony density that can be explained by a linear regression model with rock surface area as the predictor.

(c) The second observation has a very extreme y value (in the full data set consisting of 72 observations, there were two of these). This observation may have had a substantial impact on the fit of the model and subsequent conclusions. Eliminate it and recalculate the equation of the estimated regression line. (Round your values to three decimal places.)

y =

Answer 1

from above:

y^ =-298.881+9.963x

for x=70

predicted val=-298.881+70*9.963=

399

residual =20-399=-397

for x=71

predicted val=-298.881+71*9.963=

408.492 ~ 408

residual =1936-408=1528

The residual for the first point is negative, while the residual for the second point is positive.

b)

SST =                   'Σ(Yi-Y̅)2                 =		3310340.933
SSR =                   'Σ(Ŷ-Y̅)2                 =		409533.571
SSE =                   'Σ(Yi-Ŷ)2                 =		2900807.3626
Coeffficient of determination R^2 =SSR/SST=			0.124

The coefficient of determination is the proportion of the total variation in colony density that can be explained by a linear regression model with rock surface area as the predictor.

c)

y^ =41.373+0.779x