Question

Problem 1 (Modified from Problem 11-12 on page 438). An article in the Journal of Environmental En- gineering (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 (in percentage) y4.46.6 9.710.6 10.810.911.8 12.1 14.3 0.19 0.15 0.57 0.700.67 0.63 0.47 0.70 0.60 y14.715.0 17.319.223.127.427.7 31.839.5 0.78 0.81 0.780.69 1.30 1.05 1.06 1.74 1.62 (a) Plot a scatter diagram of the data. Does a simple linear regression model seem appropriate here? (b) Fit a simple linear regression model using the method of least squares, e., fit with e,"s iid N (0,02). Find β1 and A (c) Find an estimate of ơ. (d) Find the fitted value of Y corresponding to the observed data with = 0.47 and the associated (e) Estimate the mean chloride concentration for a watershed that has 2% roadway area (i.e., when residual. (Modified from Problems 11-38 and 11-56 on pages 446 and 452). Consider the data from Problem 1 in this homework on y = chloride concentration in surface streams and x-roadway area and consider the simple linear regression model in (1) (a) Test the hypothesis Ho . A-0 versus H1 : A 0 at the significance level α-5% (b) Find a 95% confidence interval on A. (c) Find a 95% confidence interval on βο (d) Find a 95% confidence interval on the m ean chloride concentration when the roadway area x-2 (e) Find a 95% prediction interval on chloride concentration when the roadway area 2-2.

Answer 1

Problem 1: a)

Scatterplot of Y vs X A0 30 20 10- 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

FROM ABOVE SCATTER PLOT IT IS SEEN THAT THERE IS POSITIVE CORRELATION BETWEEN Y AND X. Hence simple linear regression model is appropriate for this data.

b) Using Minitab and fitting simple linear regression model we have an output as

Regression Analysis: Y versus X Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value 1 1273.54 1273.54 92.22 0.000 1 1273.54 1273.54 92.22 0.000 Regression Error 16 220.95 13.81 Lack-of-Fit 14 216.4415.46 2.25 6.86 0.134 4.51 17 1494.49 Pure Error 2 Total Model Summary S R-Sq R-Sq (adj) R-sq(pred) 79.29% 3.71607 85.22% 84 , 29% Coefficients Term Constant .47 Coef SE Coef T-Value P-value VIF 1.94 2.14 0.24 0.811 9.60 0.000 1.00 20.57 Regression Equation Y 0.47 + 20.57 x

here estimates of parameters are

β,-0.47 and -20.57 c) estimate of o is is |RSS yl3.81 3.7162 d) when x-0.47 then fitted value of y and corresponding residual is 0.47 10.14 and observed value is 11.8. therefore residual is e 1.66 e) Estimate of Y when x=2 x-2 41.61

Problem 2:

a) for testing the hypothesis that there is there is no effect of X on Y consider the following output

Term Constant 0.47 1.94 0.240.811 Coef SE Coef T-Value P-Value VIF 20.57 2.149.60 0.000 1.00

The p-value of testing β1-0 is 0 and hence we conclude that X is highly significantly affected on Y

95% CI for β0 and β1 are Lower Limit Upper Limit For β0 4.574419994 3.633486302 16.0271345 25.10747567 95% CI and PI for Y when x=2 is given by Regression Equation 0.47 + 20.57 x Variable Setting 2 Fit SE Fit 2.70279 95% PI 41.6051 (35.8754, 47.3347) (31.8641, 51.3461)