Worksheet 7 PDF

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MATH2048 -- Statistics Worksheet 7 – Regression Most of this worksheet can be answered using Minitab commands, but you will also need to do some hand calculation. 1. Tungsten steel erosion shields are fitted to the low-pressure blades in steam turbines. Resistance to wear is the most important feature of these shields. Assessing the abrasion loss due to wear is not only expensive but is also destructive of the shield. However, it is expected that the abrasion loss is associated with the hardness of the steel, which can easily be measured in a nondestructive way. To investigate this relationship, the hardness of 25 shields was assessed by the Vickers method, which involves measuring the indentation left in the surface by a diamond cut in the shape of a pyramid. They were also subjected to abrasion loss tests. The data are in the file shield.txt on Blackboard (C1 = shield number, C2 = Vickers hardness, C3 = abrasion loss). Make a scatter plot. Calculate the squared correlation coefficient r2, and interpret this value. Find the least squares regression line and calculate a 95% confidence interval for the gradient. Use Minitab to generate a 95% confidence interval for (700), the mean abrasion loss for a shield with hardness equal to 700. Repeat this calculation for a shield with hardness 750 and explain why it is sensible for this confidence interval to be wider than in the first case. (Predictions in Minitab are obtained from Stat>Regression>Regression: click Options and enter an x-value in the box labelled ‘Prediction intervals for new observations’.) Use residual plots to check any assumptions that you have made in calculating the confidence intervals. 2. Tensile strength is a key property of plastic sheeting. The sheeting is made by extruding a soft plastic strip and stretching it to the required thickness. In a study of a particular machine, the rate of stretching was varied and the resulting tensile strength was measured. The data are in the file stretch.txt. C1 is the rate of stretching and C2 is the tensile strength. Make a scatter plot. Calculate the squared correlation coefficient, and interpret this value. Fit a quadratic regression equation relating tensile strength to the rate of stretching. Is the regression coefficient of X2 significant at the 5% level? The regression coefficient of X2 is negative. Assuming that the purpose of the study was to maximise the tensile strength of the sheeting, what is the practical significance of this negative coefficient? Estimate (by eye) the rate of stretching that will maximise tensile strength. Would a cubic model be a significant improvement over the quadratic?

3. The data in the file calcium.txt are from a chemical process that was run experimentally at a range of different temperatures. C1 is the temperature, C2 is the percentage calcium in the salt stage of the process and C3 is the percentage calcium in the final (metal) phase. Calculate the best fit (least squares) linear prediction equation relating the percentage calcium in the final phase (Y) to the temperature (X1) and the percentage calcium in the salt stage (X2). Calculate 95% confidence intervals for the regression coefficients of X1 and X2, and test their significance. Predict Y when X1 = 600 and X2 = 3, giving a 95% confidence interval for the prediction (the estimated mean of Y) and a 95% prediction interval for a new observation at these settings. Examine the residual plots and note any features that would cause you concern when analysing these data....