Data on the water quality in the eastern United States were obtained by a researcher who wanted to ascertain whether the amount of particulates in water (ppm) could be used to accurately predict the water quality score. Suppose we use the following simple linear regression model:

Quality_i = α+ β× particulates_i + Ɛ_i
Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software based on a sample of size 61:
$\begin{array} { | l | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 6.214 & 1.003 \\\text { Particulates } & - 0.009 & 0.020 \\\hline\end{array}$ r² = 0.005, s = 0.7896.
Is there strong evidence (and if so, why?) that there is a straight-line dependence between the amount of particulates and water quality?

Question

Data on the water quality in the eastern United States were obtained by a researcher who wanted to ascertain whether the amount of particulates in water (ppm) could be used to accurately predict the water quality score. Suppose we use the following simple linear regression model:

Quality_i = α+ β× particulates_i + Ɛ_i
Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software based on a sample of size 61:

\begin{array} { | l | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 6.214 & 1.003 \\\text { Particulates } & - 0.009 & 0.020 \\\hline\end{array}

r² = 0.005, s = 0.7896.
Is there strong evidence (and if so, why?) that there is a straight-line dependence between the amount of particulates and water quality?

Quizplus · Accepted Answer

The evidence for a straight-line dependence between the amount of particulates and water quality is considered strong if the confidence interval for the slope does not include zero. In this case, given the slope estimate (-0.009) and its standard error (0.020), and without a specific confidence level stated, the fact that the question suggests the confidence interval includes zero (as implied by the options) means there is not strong evidence of a linear relationship. The slope's standard error and the estimate itself suggest that the true slope could be zero, indicating no linear relationship.

Data on the Water Quality in the Eastern United States