Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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4. A quality control engineer has measured the num- bers of defectives per day from a certain production process for 50 days and recorded below. Test if the number of defectives follows a binomial distribu- tion at α = 0 . 05 . (20pts). number of defects frequencies 0 10 1 24 2 10 3 6 Solution. We test if data follow Binomial (3 , p ) . First we estimate p by matching the sample mean and the population mean, which gives the maxi- mum likelihood estimation of p . ¯ x = (0 · 10 + 1 · 24 + 2 · 10 + 3 · 6) / 50 = 1 . 24 = 3 p . ˆ p = 0 . 41 . Let’s see if data follow P ( X = x ) = ( 3 x ) ˆ p x (1 - ˆ p ) 3 - x . The null hypothesis we need to test is p x = P ( X = x ) , H 0 : p 0 = (1 - ˆ p ) 3 = 0 . 20 , p 1 = 3ˆ p (1 - ˆ p ) 2 = 0 . 41 , p 2 = p 2 (1 - ˆ p ) = 0 . 30 , p 3 = ˆ p 3 = 0 . 07 . The expected numbers of defects are 10.1, 20.7, 15, 3.5. Then χ 2 = (10 - 10 . 1) 2 / 10 . 1 + (24 - 20 . 7) 2 / 20 . 7 + (10 - 15) 2 / 15 + (6 - 3 . 5) 2 / 3 . 5 = 3 . 97 . The 95% cut-off value for χ 2 4 - 1 - 1 is 5.99 so we do not reject H 0 . 5. Problem on the coefficient of determination. (a) The following it the R output of a regres- sion analysis based on linear model Y = β 0 + β 1 x + for 11 paired data ( x i , y i ) . Com- pute the coefficient of determination and the coefficient of correlation (5pts). (b) It is shown during the lecture that the coef- ficient of determination is always between 0 and 1. Prove it (15pts). Solution. (a) Note that t -value = 3 = r 11 - 2 / 1 - r 2 . Solving this we get r 2 = 0 . 5 . The correlation is then r = 0 . 5 .

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