4. A quality control engineer has measured the num-bers of defectives per day from a certain productionprocess for 50 days and recorded below. Test if thenumber of defectives follows a binomial distribu-tion atα= 0.05. (20pts).number of defectsfrequencies01012421036Solution.We test if data follow Binomial(3, p).First we estimatepby matching the sample meanand the population mean, which gives the maxi-mum likelihood estimation ofp.¯x= (0·10 +1·24 + 2·10 + 3·6)/50=1.24=3p.ˆp=0.41.Let’s see if data followP(X=x) =(3x)ˆpx(1-ˆp)3-x. The null hypothesis weneed to test ispx=P(X=x),H0:p0=(1-ˆp)3= 0.20, p1= 3ˆp(1-ˆp)2= 0.41, p2=3ˆp2(1-ˆp) = 0.30, p3= ˆp3= 0.07. The expectednumbers of defects are 10.1, 20.7, 15, 3.5. Thenχ2= (10-10.1)2/10.1 + (24-20.7)2/20.7 +(10-15)2/15 + (6-3.5)2/3.5 = 3.97.The95%cut-off value forχ24-1-1is 5.99 so we do not rejectH0.5. Problem on the coefficient of determination.(a) The following it the R output of a regres-sion analysis based on linear modelY=β0+β1x+for 11 paired data(xi, yi). Com-pute the coefficient of determination and thecoefficient of correlation (5pts).(b) It is shown during the lecture that the coef-ficient of determination is always between 0and 1. Prove it (15pts).Solution.(a) Note thatt-value=3=r√11-2/√1-r2.Solving this we getr2= 0.5. The correlation is thenr=√0.5.