MATHS 2201 Engineering Mathematics IIA Assignment 3, 2017 Due: 12:00 noon on Tuesday 28th March (Week 5) 2017. Note that for full marks, please show all working, give your matlab code and include any matlab output or plots that are needed for your answers. Please give answers to four decimal places. 1. McNeese and Klein (1992) considered particles for a product that has a required mean size of = 100 microns. A random sample of 25 particles was taken and the observed sizes for the particles are shown below. 99.6 92.1 103.8 100.5 102.7 96.9 104.7 103.2 97.5 93.8 102.7 94.9 95.3 101.6 102.8 100.9 101.5 96.7 96.8 97.8 98.3 105.8 100.6 102.3 94.9 (a) State an appropriate null and alternative hypotheses to test whether the mean particle size is the required target. [1 mark] (b) To test these hypotheses, a t-test for a single mean is to be used. One of the assumptions of the t-test is normality of the data. i. Read the data into Matlab as a single vector of length 25, and then produce an appropriate plot to check if this assumption is valid. [1 mark] ii. What is your conclusion about the validity of the assumption? [1 mark] iii. What is the other assumption of the t-test? [1 mark] (c) Use the ttest function to test the hypothesis from (a) at the 5% level of significance and also obtain a 95% confidence interval for the mean particle size. Based on the Matlab output: i. state the value of the test statistic, [1 mark] ii. the degrees of freedom, [1 mark] iii. the P-value, and [1 mark] iv. the upper and lower limits for the confidence interval. [1 mark] v. Explain whether the null hypothesis should be accepted or rejected and state your final conclusion. [2 marks] [Total: 10] 2. In a certain bio-engineering experiment, a successful outcome was achieved 125 times out of 150 attempts. (a) Construct a 95% confidence interval for the probability, p, of success in a single trial. [3 marks] (b) The researchers expected a successful outcome 70% of the time. Is the data consistent with this hypothesis? Hint: consider the 95% confidence interval. [2 marks] [Total: 5] 3. Two companies, company A and company B, are used to supply electronic components. Of 60 components supplied by company A, 15 are defective, while of 70 components supplied by company B, 10 are defective. (a) Test the hypothesis that the defective proportion for company A is the same as the defective rate for company B at the 5% significance level. For full marks give the null and alternative hypothesis, the value of the test statistic, the observed P -value,and whether you reject or retain the null hypothesis. [3 marks] (b) Construct a 95% confidence interval for the difference between the defection probability of company A and company B. [2 marks] [Total: 5] 4. In an experiment concerned with accelerated testing of tyre rubber, the loss due to abrasion (gm per hour) and hardness (Shore units) were recorded for 30 samples. The data are available in the file rubber.xls. (a) Import the data into Matlab (b) Obtain a scatter plot of loss vs hardness and describe the relationship between loss and hardness. [2 marks] (c) Use the regstats function in Matlab to perform a linear regression of loss on hardness. Use the regstats dialog box to save the quantities beta, Fitted Values and Studentized Residuals. (d) Write down the estimated regression equation using the values in beta from (c) and use it to estimate the mean loss for specimens of hardness 60. [3 marks] (e) Obtain a scatter plot of the studentized residuals vs fitted values and also a normal probability plot of the studentized residuals. On the basis of these plots, do the assumptions of the linear regression model appear reasonable. [5 marks] [Total: 10] [[Assignment total: 30]] 2 ? ? ? ? ? 3