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."+ 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 CoefConstant-0.97230.3923-2.480.015-0.031170.01864-1.670.099d0.55680.31851.750.08418.26020.753624.230.0000.0028220.0039770.710.480S = 0.822059SourceR-Sq = 88.6R-Sq (adj) = 88.04DSMS0.000RegressionResidual Error4401.02100.25148.357651.360.68Total80452.38(a) How many observations were there in the data set?observations(b) Interpret the coefficient of multiple determination.O 8.0% of the observed variation in feed rate can be explained by the model relationship with vibration amplitude, depth of cut, surface roughness, and cutting speed as explanatory variables.O 8.0% of the observed variation in vibration amplitude can be explained by the model relationship with surface roughness, depth of cut, feed rate, and cutting speed as explanatory variables.O 88.6% of the observed variation in cutting speed can be explained by the model relationship with vibration amplitude, depth of cut, feed rate, and surface roughness as explanatory variables.O 88.6% of the observed variation in depth of cut can be explained by the model relationship with vibration amplitude, surface roughness, feed rate, and cutting speed as explanatory variables.O 8.6% of the observed variation in surface roughness can be explained by the model relationship with vibration amplitude, depth of cut, feed rate, and cutting speed as explanatory variables.(c) Carry out a test of hypotheses to decide if the model specifies a useful relationship between the response variable and at least one of the predictors. (Use a = 0.05.)State the appropriate hypotheses.O H: 8, + 8, + 8, + 8, +0H: at least one among B.,. .. B, is zeroO H,: B, -8, = B = 8, = 0H: at least one among 8, . B, is not zeroO H,i B, = 8, = 8, = B, = 0H: no 8 =0O H,: B, + 8, + , +8, *0H:B, = B, = B, = 8, = 0State the appropriate test statistic to two decimal places.F =What can be said about the P-value for the test?O P-value > 0.100O 0.050 < P-value < 0.100O 0.010 < P-value < 0.050O 0.001 < P-value < 0.010O P-value < 0.001State the conclusion in the problem context.O Fail to reject H. The model is judged useful.O Fail to reject H. The model is judged not useful.O Reject H. The model is judged useful.O Reject H. The model is judged not useful.

There are 4 independent variables which are s, d, f and v. The dependent variable is surface roughness. This is multiple regression model. We have to write the interpretation of coefficient of multiple determination. Then test whether the overall regression is significant or not.(a) 81 observations. (b). 88.6% of the observed variation in surface roughness can be explained by the model relationship with vibration amplitude depth of cut, feed rate, and cutting speed as explanatory variables.The null and alternative hypotheses are The value of the test statistic is F = 148.35. P-value range is P-value < 0.001. Reject H0. The model is judged useful.

Math

Statistics

The accompanying Minitab regression output is based on dara that appeared in the artice "Application of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning." The response variable is surface roughness (um), and the independent variables are vibration amplitude (um), depth of cut (mm), feed rae (mm/rev), and cuming speed (m/min), respectively. The segression equation s Ra- 0.972 - 0.0312a + 0.5574 + 18.3 + 0.00aeav Coef -0.9723 -0.03117 Predictor Constant SE Coet 0.3923 -2.40 0.016 0.0144 -.67 0.099 0.5568 0.310s 1.75 0.084 18.2402 0.002822 0.1634 0.003977 24.23 0.000 0.71 0.480 S- 0.822059 Source Regression Residual Error R-Sgadg)- 88.04 DS 100.a6 401.02 51.34 140.35 0.000 76 0.60 Total 80 452.38 (a) How many observations were there in the data ser? observations (b) Interpret the coefficient of multiple determination. O 88.0% of the observed variation in feed rate can be explained by the model relationship with vibration amplitude. depth of cut surface roughness, and cuting speed as explanatory variables. O 88.0 of the observed variation in vibration amplitude can be explained by the model relationship with surface roughness. depth of out feed rate. and cutting speed as eplanatory variables. O 88.6 of the observed variation in cutting speed can be eplained by the model relationship with vibration amplitude. depth of out. feed rate. and surface roughness as eplanatory variables. O 88.6% of the observed variation in depeh of cut can be eplained by the model relationship with vibration amplitude. surface roughness. feed rate. and cuting speed as eplanatory variables. O 88.6% of the observed variation in surface roughness can be explained by the model relationship with vibration amplitude. depth of out feed rate and cutting speed as explanatory variables. (e) Camy out a test of hypotheses to decide if the model specifies a useful relationship bereen the response variable and at least one of the predictors. (Use a 0.05.) State the appropriate hypotheses OH, ,, 0 at least one among m, is zero OH:P , .-0 H least one ameng , not zero OH,,,,0 OH,,,.0 H ,,0 - State the appropriate test statistic to twe decimal places. what can be said about the Prvalue for the test? O Palue > 0.100 O 0.050 Pvalue <0.100 O 0.010 < Pralue < 0.050 O 0.001 Palue 0.010 OAvalue <0.001 State the condlusion in the problem contet O Fail to reject H, The model is judged useful. O Fail to reject H The model is judged not useful. O Regect H The model a judged usefu. O Reject H. The model s judged not useful.