using r- studio please cod the following parts a-h =pri-6-46. The data shown in Table 6-15 represent the thrust of ajet-turbine engine (y) and six candidate regressors: x₁mary speed of rotation, x₂ = secondary speed of rotation, x3fuel flow rate, x = pressure, x, = exhaust temperature, andx6 = ambient temperature at time of test.(a) Fit a multiple linear regression model using x, = fuel flowrate, x4 pressure, and x5 exhaust temperature as the==regressors.(b) Test for significance of regression using a = 0.01. Findthe P-value for this test. What are your conclusions?(c) Find the t-test statistic for each regressor. Using a = 0.01,explain carefully the conclusion you can draw from thesestatistics.(d) Find R2 and the adjusted statistic for this model. Commenton the meaning of each value and its usefulness in assess-ing the model.(e) Construct a normal probability plot of the residuals andinterpret this graph.(f) Plot the residuals versus y. Are there any indications ofinequality of variance or nonlinearity?(g) Plot the residuals versus x3. Is there any indication of non-linearity?(h) Predict the thrust for an engine for which x3-X4 = 170, and x = 1589.= 20000, YX4 X5 X64540 2140 20640 30250 205 1732 99X1 X2X3نت4315 2016 20280 30010 195 1697 1004095 1905 1986029780 184 1662 973650 1675 1898029330 164 1598 973200 1474 18100 28960 144 1541 974833 2239 20740 30083 215 1709 874617 2120 20305 29831 206 1669 874340 1990 19961 29604 195 1640 873820 1702 18916 29088 171 1572 853368 1487 18012 28675 149 1522 854445 2107 20520 30120 195 1740 1014188 1973 20130 29920 190 1711 1003981 1864 19780 29720 180 1682 1003622 1674 19020 29370 161 1630 1003125 1440 180301572 1014560 2165 20680 30160208 1704 984340 2048 20340 29960 1991679 964115 1916 1986029710 187 1642 943630 1658 1895029250 164 1576 943210 1489 1870028890 145 1528 944330 2062 2050030190 193 1748 1014119 1929 20050 29960 183 1713 1003891 1815 19680 29770 173 1684 1003467 1595 18890 29360 153 1624 993045 1400 17870 28960 134 1569 1004411 2047 20540 30160 193 1746 994203 1935 20160 29940 184 1714 993968 1807 19750 29760 173 1679 9928940 1393531 1591 18890 29350 153 1621 993074 1388 17870 28910 133 1561 994350 2071 20460 30180 198 1729 1024128 1944 20010 29940 186 1692 1013940 1830 19640 29750 178 1667 1013480 1612 18710 29360 156 1609 1013064 1410 17780 28900 136 1552 1014402 2066 20520 30170 197 1758 1004180 1954 20150 29950 188 1729 993973 1835 19750 29740 178 1690 993530 1616 18850 29320 156 1616 993080 1407 17910 28910 137 1569 100

The dataset contains observationsDependent variable:: Thrust of jet, turbine engine.Independent variables (Regressors):: Primary speed of rotation.: Secondary speed of rotation.: Fuel flow rate.: Pressure.: Exhaust temperature.: Ambient temperature.The R-code for fitting of the multiple linear regression model:1# Given data 2 3y=c(4540,4315,4095,3650,3200,4833,4617,4340,3820,3368,4445,4188,3981,3622,3125,4560,4340,4114,3630,3210,4330,4119,3891,3467,3045,4411,4203,3968,3531,3074,4350,4128,3940,3480,3064,4402,4180,3973,3530,3080) 4 5X3=c(30250,30010,29780,29330,28960,30083,29831,29604,29088,28675,30120,29920,29720,29370,28940,30160,29960,29710,29250,28890,30190,29960,29770,29360,28960,30160,29940,29760,29350,28910,30180,29940,29750,29360,28900,30170,29950,29740,29320,28910) 6 7X4=c(205,195,184,164,144,215,206,195,171,149,195,190,180,161,139,208,199,187,164,145,193,183,173,153,134,193,184,173,153,133,198,186,178,156,136,197,188,178,156,137) 8 9X5=c(1732,1697,1662,1598,1541,1709,1669,1640,1572,1522,1740,1711,1682,1630,1572,1704,1679,1642,1576,1528,1748,1713,1684,1624,1569,1746,1714,1679,1621,1561,1729,1692,1667,1609,1552,1758,1729,1690,1616,1569) 10 11# (a) Fit a multiple linear regression model using X3, X4, X5 as regressors. 12 13model=lm(y ~ X3 + X4 + X5) 14summary(model)The output of the code:> summary(model)Call:lm(formula = y ~ X3 + X4 + X5)Residuals: Min 1Q Median 3Q Max -77.340 -26.631 -5.083 14.481 85.014 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4210.6811 2327.5427 1.809 0.07880 . X3 -0.2313 0.1119 -2.068 0.04591 * X4 21.4846 0.9381 22.903 < 2e-16 ***X5 1.6916 0.5633 3.003 0.00484 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 44.63 on 36 degrees of freedomMultiple R-squared: 0.9928, Adjusted R-squared: 0.9922 F-statistic: 1650 on 3 and 36 DF, p-value: < 2.2e-16Therefore the multiple linear regression equation for the model using as regressors is,.The R-code for the test for significance of regression using .1# (b) Test for significance of regression using alpha=0.01. 2 3# Find the p-value for this test. 4 5anova_result=anova(model) 6 7p_value=anova_result$`Pr(>F)`[1] 8 9# Display the p-value 10 11print(p_value) 12 13# Check if the p-value is less than alpha=0.01 for significance 14 15if (p_value < 0.01) { 16 17cat("The regression model is significant.\n") 18 19} else { 20 21cat("The regression model is not significant.\n") 22 23} 24 25The output of above code-> print(p_value)[1] 4.618411e-39> # Check if the p-value is less than alpha=0.01 for significance> if (p_value < 0.01) {+ cat("The regression model is significant.\n")+ } else {+ cat("The regression model is not significant.\n")+ }The regression model is significant.The p-value of the test is therefore the regression model is significant.The R-code for finding the t-test statistics for each regressor1# (c) Find the t-test statistic for each regressor. 2 3summary_result=summary(model) 4 5# Extract t-test statistics and p-values 6 7t_stats = summary_result$coefficients[, "t value"] 8 9p_values=summary_result$coefficients[, "Pr(>|t|)"] 10 11# Display t-test statistics and p-values 12 13print(data.frame(Regressor = rownames(summary_result$coefficients), t_stats, p_values)) 14 15# Check significance using alpha=0.01 16 17significant_regressors=names(t_stats[p_values < 0.01]) 18 19cat("Significant regressors:", paste(significant_regressors, collapse = ", "), "\n")The output of the above Code-> print(data.frame(Regressor = rownames(summary_result$coefficients), t_stats, p_values)) Regressor(Intercept) (Intercept)X3 X3X4 X4X5 X5 t_stats(Intercept) 1.809067X3 -2.067706X4 22.902840X5 3.003081 p_values(Intercept) 7.879793e-02X3 4.591245e-02X4 4.709029e-23X5 4.837918e-03> cat("Significant regressors:", paste(significant_regressors, collapse = ", "), "\n")Significant regressors: X4, X5 The pressure and Exhaust temperature are the significant regressors.The R-code for the and the adjusted :1# (d) Find R-square and adjusted R-square for this model. 2 3rsquare=summary_result$r.squared 4 5adjusted_rsquare=summary_result$adj.r.squared 6 7# Display R-square and adjusted R-square 8 9cat("R-square:", rsquare, "\n") 10 11cat("Adjusted R-square:", adjusted_rsquare, "\n")The output of the above code-> cat("R-square:", rsquare, "\n")R-square: 0.9927812> cat("Adjusted R-square:", adjusted_rsquare, "\n")Adjusted R-square: 0.9921796Interpretation of both these values:The value of is and the value of adjusted is .The of means that approximately 99.28% of the variability in the thrust of jet turbine engine can be explained by the variation in fuel flow rate, pressure, and exhaust temperature included in the model.Adjusted is modified version of R Square that accounts for the number of predictors in the model. It penalized the model for including irrelevant predictors that do not significantly improve the fit. An adjusted of means that, after adjusting for the number of predictors, approximately 99.22% of the variability in dependent variable is still explained by the model.The R-code for the construction of the Normal probability plot of the model,1# (e) Construct a normal probability plot of the residuals and interpret this graph. 2residuals = resid(model) 3 4# Set up a new graphics window with a specified size 5options(repr.plot.width=4, repr.plot.height=4) # Adjust the width and height as needed 6 7# Create the normal probability plot 8qqnorm(residuals) 9qqline(residuals) 10 11# Reset the graphics options to default 12options(repr.plot.width=NULL, repr.plot.height=NULL)The normal probability plot of the residual:Interpretation of the graph-The points fall approximately on a straight line therefore the residuals are normally distributed.The R-code for the plot of the residuals versus 1# (f) Plot the residuals versus yhat. Are there any indications of inequality of variances or nonlinearity? 2yhat=predict(model) 3plot(yhat, residuals) 4abline(h = 0, col = "red", lty = 2)The output of the R-code-The plot of residuals versus Interpretation of the plot of the residual versus fitted value plot.Since the spread of residencies is roughly constant across predicted values, the assumption of equal variance is met.The R-code for the plot of residual versus :1# (g) Plot the residuals versus X3. Is there any indication of nonlinearity? 2plot(X3, residuals) 3abline(h = 0, col = "red", lty = 2) # Interpretation: If there's a clear pattern in the residuals against X3, there might be nonlinearity.The output of the above R-code:The plot of the residuals versus Interpretation:Since there is no clear pattern in the plot the data is linear.The R-code for predicting the thrust for the given values of the regressor variable:1# (h) Predict the thrust on engine for which X3=20000, X4=170, X5=1589 2new_data = data.frame(X3 = 20000, X4 = 170, X5 = 1589) 3prediction = predict(model, newdata = new_data) 4 5# Display the predicted thrust 6cat("Predicted Thrust:", prediction, "\n")The output of the above code:> cat("Predicted Thrust:", prediction, "\n")Predicted Thrust: 5924.175 The thrust for an engine for which is .(a) Therefore the multiple linear regression equation for the model using as regressors is,.(b) The p-value of the test is therefore the regression model is significant.(c) The pressure and Exhaust temperature are the significant regressors.(d) The value of is and the value of adjusted is .The of means that approximately 99.28% of the variability in the thrust of jet turbine engines can be explained by the variation in fuel flow rate, pressure, and exhaust temperature included in the model.Adjusted is modified version of R Square that accounts for the number of predictors in the model. It penalized the model for including irrelevant predictors that do not significantly improve the fit. An adjusted of means that, after adjusting for the number of predictors, approximately 99.22% of the variability in dependent variable is still explained by the model.(e) The normal probability plot of the residual:The points fall approximately on a straight line therefore the residuals are normally distributed.(f) The plot of residual versus :Since the spread of residencies is roughly constant across predicted values, the assumption of equal variance is met.(g) Plot of the residual versus :Since there is no clear pattern in the plot the data is linear.(h) The thrust for an engine for which is .