The accompanying Minitab regression output is based on data that appeared in the article "Application of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning."t The response variable is surface roughness (um), and the independent variables are vibration amplitude (um), depth of cut (mm), feed rate (mm/rev), and cutting speed(m/min), respectively.The regression equation isRa = -0.972 - 0.0312a + 0.557d + 18.3f + 0.00282vPredictorCoefSE CoefPConstant-0.97230.3923-2.480.015a-0.031170.01864-1.670.099d.0.55680.31851.750.08418.26020.002822R-Sq = 88.6%f0.753624.230.0000.0039770.710.480S = 0.822059R-Sq (adj) = 88.0%SourceDSMSRegression4401.02100.25148.350.000Residual Error7651.360.68Total80452.38 (d) Interpret the number 18.2602 that appears in the Coef column.O This is the coefficient on v, after adjusting for the effects of the other variables, a 1 m/min increase in cutting speed is associated with an estimated decrease of 18.2602 um in expected surface roughness.O This is the coefficient on d, after adjusting for the effects of the other variables, a 1 mm increase in depth of cut is associated with an estimated increase of 18.2602 um in expected surface roughness.O This is the coefficient of the constant, after adjusting for the effects of the other variables, a 1 µm increase in feed rate is associated with an estimated increase of 18.2602 pm in expected surface roughness.O This is the coefficient on a, after adjusting for the effects of the other variables, a 1 µm increasevibration amplitude is associated with an estimated decrease of 18.2602 um in expected surface roughness.O This is the coefficient on f, after adjusting for the effects of the other variables, a 1 mm/rev increase in feed rateassociated with an estimated increase of 18.2602 um in expected surface roughness.(e) At significance level 0.10, can any single one of the predictors be eliminated from the model provided that all of the other predictors are retained? (Select all that apply.)O vibration amplitudeO depth of cutO feed rateO cutting speedO surface roughness(f) The estimated SD of Y when the values of the four predictors are 10, 0.5, 0.25, and 50, respectively, is 0.1174. Calculate both a 95% CI for true average roughness and a 95% PI for the roughness of a single specimen. (Round your answers to two decimal places.)confidence intervalUmprediction interval|UmCompare these two intervals.O The prediction interval is the same width as the confidence interval.O The prediction interval is much wider than the the confidence interval.O The prediction interval is much narrower than the the confidence interval.

Given information: The MINITAB output for multiple linear regression is given.(d). Interpret the number 18.2602: Multiple linear regression model: A multiple linear regression model is given as y = b0 + b1x1 + b2x2 + …+ bkxk where y is the predicted value of response variable, and x1, x2,…, xk are the k predictor variables. The quantities b1, b2,…, bk are the estimated slopes corresponding to x1, x2,…, xk respectively and b0 is the estimated intercept of the line, from the sample data. A multiple regression equation describes the combined effect of all the predictors in the model. Even when the effect of a particular predictor is being studied from a multiple regression equation, it assumes that the effects of all the other predictors are accounted for. From the accompanying MINITAB output it is seen that, the value 18.2602 is the regression coefficient of the predictor variable f. Assuming that the effects of all the other predictors are accounted for, the value of response variable increases by 18.2602 units for one unit increase in the variable f. This is the coefficient on f, after adjusting for the effects of the other variables, a 1 mm/rev increase in feed rate is associated with an increase of 18.2602 µm is expected surface roughness.(e). Test for the overall significance of the model: The investigator is interested to test whether the regression predictor variables are significant or not. Denote βi as the regression coefficient for each of the predictor variable i. The null and alternate hypotheses are stated below: Null hypothesis H0: H0: βi = 0 That is, the predictor variable i is not statistically significant. Alternative hypothesis H1: H1: βi ≠ 0 That is, the predictor variable i is statistically significant. The level of significance is α = 0.10. Decision rule based on P-value approach: If P-value ≤ α, then reject the null hypothesis H0. If P-value > α, then fail to reject the null hypothesis H0. From the accompanying regression output, the P-values of each of the predictor variables and their significance is as shown below: Predictor variable P-value Comparison with α Significance Vibration amplitude 0.099 0.099 < 0.1 Significant Depth of cut 0.084 0.084 < 0.1 Significant Feed rate 0.000 0.000 < 0.1 Significant Cutting speed 0.71 0.71 > 0.1 Not significant Thus, the predictor variable Cutting speed can be eliminated from the regression model.(f). Calculate the 95% confidence interval for the true average roughness for given values: The estimated standard deviation of the predicted y when x1 = 10, x2 = 0.5, x3 = 0.25 and x4 = 50 is 0.1174. The degrees of freedom is d.f = n – k – 1 = 81 – 4 – 1 = 76. The critical value in t-distribution corresponding to α = 0.05, and df = 76 is “= T.INV.2T(0.05, 76)” is 1.9917. The predicted value of y when x1 = 10, x2 = 0.5, x3 = 0.25 and x4 = 50 is obtained as given below: y^=-0.9723-0.03117a+0.5568d+18.2602f+0.002822v=-0.9723-0.03117×10+0.5568×0.5+18.2602×0.25+0.002822×50=-0.9723-0.3117+0.2784+4.56505+0.1411=3.70055 The 95% confidence interval for the true average roughness for given values is obtained as given below: CI=y^-tα2,df×sey^, y^+tα2,df×sey^=3.70055-1.9917×0.1174, 3.70055+1.9917×0.1174=3.70055-0.23382558, 3.70055+0.23382558=3.47, 3.93 The 95% prediction interval for the true average roughness for given values is obtained as given below: CI=y^-tα2,df×MSE+sey^2,y^+tα2,df×MSE+sey^2y^=3.70055-1.9917×0.68+0.11742, 3.70055-1.9917×0.68+0.11742=3.70055-1.658959, 3.70055+1.658959=2.04, 5.36 The prediction interval is much wider than the confidence interval.