3. Mechanical micro-cutting is emerging as a viable alternative to lithography based micromachin- ing techniques. A new laser assisted mechanical micromachining [LAMB-'1) process has been proposed in which a laser is used to soften the material before cutting. A study examined the accuracy of the groove cut using this approach. The LAMM process was used to cut grooves to a nominal depth of 20 pm in 10 specimens of H13 steel. The actual depth of the groove cut using the LAMM process was measured using white light interferometry. From the 10 specimens, the sample mean of the measured depth of the grooves was 19 pm. The population standard deviation can be assumed to be 2 pm. (a) In parts (1)) - {e} we will construct condence intervals and perform hypothesis tests based on the collected data. What assumptions are needed to apply these methods? The data was stored in the array Lam and a plot (below) generated in python. Based on this plot, is the data consistent with these assumptions? [4 marks] import matplotlib.pyplot as plt import scipy.stats as stats stats . probplot (LMlM , dist='norm' ,plot=plt) p1t.show() Probability Plot 23' 21' 20~ Ordered Values 15' 1?- l l' I' |' I 1.5 1.0 05 00 05 10 15 Theoretical quantiles (h) Construct a 95% condence interval for the expected groove depth. [3 marks] (c) How many samples would be required for the margin of error of the 99% condence interval for the expected groove depth to be no greater than 0.5 Jim? [3 marks] Ed} Does the study give any evidence of a bias in depth of the groove. In other words. is there any evidence that the expected depth of the groove is different from the nominal value of 20 ,um. State the null and alternative hypotheses. Carry out an appropriate test {with population standard deviation assumed known} to determine the P-value. What do you conclude? [6 marks] (e) Suppose the true expected depth of the groove is 18 pm. Determine the power of the test in part {d} to detect this difference at the 5% signicance level. [5 marks]