Stat 2118 - Homework 1 Question 1 In a study, the protein absorption (Y) for seven concentration levels (X) of that protein were measured: Conc. Level (Xi) 6 8 10 12 14 16 18 Absorption (Yi) 10 15 18 18 24 22 26 a) Find the least squares estimate for the regression line Yi = 0 + 1Xi + i. b) What would be your estimate of the absorption when the concentration level is 10? c) Estimate the standard deviation of the error term i d) Compute the correlation coefficient. e) What proportion of the variability in the absorption can be explained using this model (i.e. what is R2)? f) Test the null hypothesis that 1 = 0 (perform a two-sided test), using = 0.05. Is the model useful? Question 2 With the growth of internet service providers, a researcher decides to examine whether there is a correlation between cost of internet service per month (rounded to the nearest dollar) and degree of customer satisfaction (on a scale of 1 - 10 with a 1 being not at all satisfied and a 10 being extremely satisfied). The researcher only includes programs with comparable types of services. A sample of the data is provided below. dollars satisfaction 11 6 18 8 17 10 15 4 9 9 5 6 12 3 19 5 22 2 25 10 a) Find the equation of the fitted regression line. b) Estimate the standard deviation of the error term i c) Compute the correlation coefficient c) What is the R2 for this model? e) Calculate a 90% confidence interval for 1 f) Test the null hypothesis that 1 = 0 (perform a two-sided test), using = 0.1. Is the model useful? Charissa Santoso STAT 2118 Homework 1 Question 1 Total Mean of Xi Mean of Yi Concentration Level Xi Absorption (Yi) 6 8 10 12 14 16 18 84 (Xi)^2 (Yi)^2 Yi(hat) (Yi - Yi Hat)^2 (Xi)(Yi) Xi - Mean of Xi Yi - Mean of Yi (Xi - Mean of Xi) x (Yi - Mean of Yi) (Xi- Mean of Xi) ^2 (Yi - Mean of Yi)^2 36 100 11.716 2.944656 60 -6 -9 54 36 81 64 225 14.144 0.732736 120 -4 -4 16 16 16 100 324 16.572 2.039184 180 -2 -1 2 4 1 144 324 19 1 216 0 -1 0 0 1 196 576 21.428 6.615184 336 2 5 10 4 25 256 484 23.856 3.444736 352 4 3 12 16 9 324 676 26.284 0.080656 468 6 7 42 36 49 1120 2709 133 16.857152 1732 0 0 136 112 182 10 15 18 18 24 22 26 133 12 19 b1 = Sum of (Xi - Mean of X).(Yi- Mean of Yi) Divide By Sum of ( Xi - Mean of X)^2 b1=136/112 = 1.214 b0 = Mean of Y - b1 times mean of X b0= 19 - 1.214 (12) = 4.432 1.21 a) Least square estimate for the regression line is = Yi = b0 + b1X Yi (hat)= 4.432 + 1.214 x b) If X =10, Yi = 4.432 + 12.14 = 16.572 The estimate is 16.5714 , given if Concentration Level = 10 c) Variance = SSE / n-2 = 16.857 / (7 - 2) = 3.3714 Standard Deviation = Square Root of 3.3714 = 1.836 = 1.84 d) 3.3714304 16.572857 1.2142857143 4.4285714286 16.5714285714 0.9074 0.953 0.907 1.836145528 2.6457513 test statistic 17.2911 12 0.6939978 2.645751311 17.291122 27.377609 Question 2 Total Mean of Xi Mean of Yi Dollars ( Xi ) (Xi)^2 (Yi)^2 6 121 8 324 10 289 4 225 9 81 6 25 3 144 5 361 2 484 10 625 63 2679 Satisfaction (Yi) 11 18 17 15 9 5 12 19 22 25 153 Yi(hat) (Yi - Yi Hat)^2 36 6.1484 0.022034465 64 6.4004 2.558591853 100 6.3644 13.21729574 16 6.2924 5.255281642 81 6.0764 8.547202451 36 5.9324 0.004564339 9 6.1844 10.14065879 25 6.4364 2.063360179 4 6.5444 20.65193588 100 6.6524 11.20615724 471 63.032 73.66708258 (Xi)(Yi) Xi - Mean of Xi Yi - Mean of Yi (Xi - Mean of Xi) x (Yi - Mean of Yi) (Xi- Mean of Xi) ^2 (Yi - Mean of Yi)^2 66 -4.3 -0.3 1.29 18.49 0.09 144 2.7 1.7 4.59 7.29 2.89 170 1.7 3.7 6.29 2.89 13.69 60 -0.3 -2.3 0.69 0.09 5.29 81 -6.3 2.7 -17.01 39.69 7.29 30 -10.3 -0.3 3.09 106.09 0.09 36 -3.3 -3.3 10.89 10.89 10.89 95 3.7 -1.3 -4.81 13.69 1.69 44 6.7 -4.3 -28.81 44.89 18.49 250 9.7 3.7 35.89 94.09 13.69 976 -7.105427E-015 1.77636E-015 12.1 338.1 74.1 15.3 6.3 0.0357882283 a) b1 = Sum of (Xi - Mean of X).(Yi- Mean of Yi) Divide By Sum of ( Xi - Mean of X)^2 b1 = 12.1/338.1=0.0357882 = 0.036 0.0357882283 b0= Mean of Y - b1 times mean of X 5.752 b0=6.3 - 0.036 * 15.3= 5.7492 9.20839 Y(hat) = 5.7492 + 0.036x 3.03453 b) Variance = (73.669776) / (10-2) = 1/8 = 3.0345 Standard Deviation of the error term = Square root 0.125 = 0.353553 c) 12.1 / (Square Root of 338.1*74.1) = 0.0765 d) R*2 = 1 9.2083853 25053.21 d) R^2 = 1 158.28205836 3.03 0.0764458 0.076445809 0.0058 0.036