Question: Three replicated water samples were taken at each of four locations in a river to determine whether the quantity of dissolved oxygen, a measure of water pollution, varied from one location to another (the higher the level of pollution, the lower the dissolved oxygen reading). Location 1 was adjacent to the waste-water point for a certain industrial plant, and location 2, 3, and 4 were selected at points 10, 20 and 30 miles downstream from this discharge point. The resulting data appear in the accompanying table.

Use the information below to answer the following questions:

1. (3 marks) Is there a difference in mean dissolved oxygen content among the four locations? Just state the test-statistic, the p-value and answer the question.
2. (6 marks) Suppose it was conjectured that the mean dissolved oxygen content for the first location is different from other locations, because its location was adjacent to the waste-water point. Carry out a single overall test (regardless of your results in part (a)) to determine if there is a difference in mean dissolved oxygen content among the last three tree locations. In terms of parameters, state clearly the models in the null and alternative hypothesis. Also, identify the sum of squared residuals and degrees of freedom for the models in the null and alternative hypothesis. Calculate the test-statistic, the p- value and identify the distribution of the test-statistic under the null hypothesis. What do you conclude?
3. (4 marks) Suppose it was conjectured that the locations caused increase in dissolved oxygen content and that this increase accumulated as the location was farther from the discharge point. Define a linear combination to define a measure of the linear relationship between location (the four equally spaced distance 0, 10, 20, and 30 miles downstream from the plant) and the mean of dissolved oxygen content. Carry out a test about the conjecture. Interpret your conclusion.
4. (3 marks) Another way to quantify the strength of the linear relationship between the distance of location (Length) and the mean of dissolved oxygen content ( ) is to fit the regression model *Length. Fit the regression model and carry out a test for :β1 versus H1 : Interpret your conclusion.
5. (2 marks) Are p-values in parts (c) and (d) the same? Should they be? Explain your idea.

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