Question

An article in the Journal of Environmental Engineering (Vol. 115, No. 3, 1989, pp. 608–619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The following data are chloride concentration y (in milligrams per liter) and roadway area in the watershed x (in percentage).

a. Draw a scatter diagram of the data. Does a simple linear regression model seem appropriate here? Explain.
b. Test the hypothesis H0: β1 = 0. What do you determine about the slope of the regression line?
c. Fit the simple linear regression model using the method of least squares and provide the equation of the line.
d. Estimate the mean chloride concentration for a watershed that has 1% roadway area.
e. Find the fitted value corresponding to x = 0.47 and the associated residual.
f. Test H0: β0 = 0 versus H1: β0 ≠ 0 using α = 0.01. What conclusions can you draw?
g. Calculate R2 for this model. Provide an interpretation of this quantity.
h. Test the hypothesis that H0: ρ = 0 using alpha = 0.05. Interpret the results.

Answer 1

Answering the statistical question involves drawing a scatter plot, testing for a significant relationship using hypothesis testing, fitting a linear regression model, predicting y values, and assessing the goodness-of-fit with R².

Step-by-step explanation:

The correct answer involves a series of steps to understand and interpret the relationship between two variables using simple linear regression analysis. The method of scatter plots and least-squares regression line are tools for visualizing and quantifying this relationship.

To begin, one would draw a scatter plot of the data to observe the distribution and relationship patterns between the independent variable (x) and the dependent variable (y). A linear pattern in the scatter plot suggests that a simple linear regression model is appropriate.

Next, testing whether the slope (β1) is significantly different from zero (hypothesis testing) allows us to understand if there is a statistically significant linear relationship between x and y.

The least-squares regression line equation (Ŷ = a + bx) is computed to fit the model to the data, where 'a' represents the y-intercept and 'b' represents the slope of the line. The y-intercept may have a context-dependent interpretation.

Calculating the correlation coefficient (r) and determining its significance provides insight into the strength and direction of the linear relationship. If the null hypothesis that the population correlation coefficient (ρ) is zero can be rejected, it implies a significant linear relationship.

Lastly, one can use the regression line to predict y values at given x values and assess the model's goodness-of-fit through the coefficient of determination (R²), which quantifies the proportion of the total variation in y that is explained by the linear relationship with x.