An article in Wear (Vol. 152, 1992, pp. 171-181) presents data on the fretting wear of mild steel and oil viscosity. Representative datafollow, with x = oil viscosity and y = wear volume (10-4 cubic millimeters).y 240 181 193 155 172 110 113 75 941.69.4(b) Slope.[iXRound your answers to 3 decimal places.Find a 90.0% confidence interval on each of the following:(a) Intercept.[iii(c) Mean wear when oil viscosity x = 25.0.[i15.5 20.0 22.0 35.5 43.0 40.5 33.0i]]]

Let x be the oil viscosity and y denote the wear volume.The data on x and y with their respective mean are as follows yx2401.61819.419315.5155201722211035.5113437540.59433SUM∑y=1333∑x=220.5MEANy¯=148.1111x¯=24.5The defined confidence level is (1−α)=0.9Thus the significance level is α=0.1The sample size is n=9It is required to find the 90% confidence interval fora) Intercept (α)b) Slope (β)c) Mean wear when x=25Consider the following simple linear equation,To compute the coefficients consider the following table xy(xi−x¯)(yi−y¯)(xi−x¯)∗(yi−y¯)(xi−x¯)21.6240-22.9091.8889-2104.26524.419.4181-15.1032.8889-496.62228.0115.5193-9.0044.8889-404.0081.0020155-4.506.8889-31.0020.2522172-2.5023.8889-59.726.2535.511011.00-38.1111-419.22121.004311318.50-35.1111-649.56342.2540.57516.00-73.1111-1169.78256.0033948.50-54.1111-459.9472.25SUM220.51333.0000.000.0000∑(xi−x¯)∗(yi−y¯)=−5794.100∑(xi−x¯)2=1651.42Thus, the estimate of the slope term is The estimated value of the intercept coefficient is a) The formula to compute the confidence interval for the intercept is given asCI(α^)=α^ ± t(α2,n−2)∗SE(α^)Here, t(α,n−2)=t(0.1,7) is computed using the Excel function =TINV(0.1, 7)Thus the t-critical value is 1.895SE(α^) is the standard error of the intercept and is computed as followsHere, MSE is the mean squared error and is given as ∑(yi−y^i)2n−2 xiyiy^iei=(yi−y^i)ei21.6240228.45711.54295024133.23970039.4181201.0903-20.09031083403.620589315.5193179.688113.31188243177.206213920155163.8996-8.89961433979.2031353822172156.882515.11749821228.53875235.5110109.5170.483007890.2332966214311383.2028229.7971799887.871932140.57591.97421-16.97421075288.12383053394118.2884-24.28838279589.9255387∑(yi−y^i)2=2787.962989Thus, MSE=2787.9629897=398.280427Thus, the standard error of the intercept isHence the required confidence interval is Thus the 90% confidence interval for the intercept is b) The formula to compute the confidence interval for the Slope is given asCI(β^)=β^ ± t(α2,n−2)∗SE(β^)Here, The t-critical value is 1.895SE(β^) is the standard error of the slope and is computed as followsHence the required confidence interval is The 90% confidence interval for the Slope is c) The formula to compute the interval for the mean response is given as followsCI(y^)=yo^±tα/2,n−2⋅MSE(1+1n+(xo−x¯)2∑(xi−x¯)2)Here, x0=25The predicted value of y at x0 is as followsThe t-critical value is 1.895Thus the required confidence interval is The 90% confidence interval for the mean Wear when the Oil viscosity is 25 is a) The 90% confidence interval for the intercept is b) The 90% confidence interval for the Slope is c) The 90% confidence interval for the mean Wear when the Oil viscosity is 25 is