Consider a manufacturing process for making pistons from metal ingots in which each ingot produces enough material for1000
pistons. Sometimes, a piston cracks whilst cooling after being forged. Previous research has shown that, in a batch of1000
pistons, the average number of pistons that develop a crack during cooling is dependent on the purity of the ingots. Ingots of known purityX
were forged into pistons, and the average numberY
of cracked pistons per batch was recorded in the table below.Each batch is made up of a random sample of pistons of the same purity.
Consider a manufacturing process for making pistons from metal ingots in which each ingot produces enough material for 1000pistons. Sometimes, a piston cracks whilst cooling after being forged. Previous research has shown that, in a batch of 1000pistons, the average number of pistons that develop a crack during cooling is dependent on the purity of the ingots. Ingots of known purity X were forged into pistons, and the average number Y of cracked pistons per batch was recorded in the table below. Each batch is made up of a random sample of pistons of the same purity. Ingots purity I; 0.960.940.950.980.980.97 Average number y; Of cracked pistons 3.795.074.432.892.063.16 Below is the data from the table reproduced in a format which you can copy-paste easily. 0. 96 0. 94 0. 95 0. 98 0. 99 0.97 3. 79 5. 07 4. 43 2. 89 2. 06 3. 16 One has I = 0.965 and ( 1)2 = 0.00175. The following linear regression model is fit to these data: Y = Bo + BIX+E. Use the following Matlab regression output to answer the questions below. (Note that you may need to adjust the width of your browser window for the output to be displayed properly) Linear regression model: Y1 +X Estimated Coefficients: Estimate SE tStat pValue ( Intercept) 59. 537 2. 862 20. 81 3. 15e-05 X -58. 000 2. 965 -19. 56 4. 03e-05 Number of observations: 6, Error degrees of freedom: 4 Root Mean Squared Error: 0. 1240 R-squared: 0. 9897e) [5 marks] Perform a hypothesis test to determine whether the variable X is significant in this model, at the 5% level of significance Hypotheses: Ho : B1 H1 : B1 = (Enter the exact values of the numerical answers) The observed test statistic is t = (Enter your answer correct to at least 2 decimal places) The test statistic follows a t distribution on degrees of freedom if Ho is true. (Enter the exact value) The p-value is (Enter your answer correct to at least 4 decimal places) Hence, we have evidence that there is an association between ingots purity and average number of cracked pistons per batch. f) [2 marks] Compute a two-sided 95% confidence interval for the parameter P1. (Enter your answer correct to at least 4 decimal places) g) Consider an ingot purity of X = 0.975. i) [1 mark] Estimate the average number of cracked pistons in a batch forged from ingots of purity X" = 0.975. (Enter your answer correct to at least 2 decimal places) ii) [2 marks] Compute a two-sided 95% confidence interval for the expectation of Y" when ingots purity is set to X = 0.975. (Enter your answer correct to at least 4 decimal places)h) [4 marks] For each of the following, determine if they are an assumption which must be satisfied in order for the regression analysis to be valid, and where appropriate, comment on the validity of the assumption by constructing an appropriate plot. The errors have been drawn from a normal distribution. Yes, it is an assumption needed and it seems (approximately) reasonable. Yes, it is an assumption needed but it does not seem reasonable here. Yes, it is an assumption needed but we don't have enough information to verify its validity. O No, it is not an assumption needed. The mean of y is a linear function of r. O Yes, it is an assumption needed and it seems (approximately) reasonable. O Yes, it is an assumption needed but it does not seem reasonable here. Yes, it is an assumption needed but we don't have enough information to verify its validity. O No, it is not an assumption needed. The errors are independent of one another. Yes, it is an assumption needed and it seems (approximately) reasonable. O Yes, it is an assumption needed but it does not seem reasonable here. O Yes, it is an assumption needed but we don't have enough information to verify its validity. O No, it is not an assumption needed. The errors have the same variance. O Yes, it is an assumption needed and it seems (approximately) reasonable. Yes, it is an assumption needed but it does not seem reasonable here. Yes, it is an assumption needed but we don't have enough information to verify its validity. O No, it is not an assumption needed
