2. Suppose that a database contains ten thousand errors. Two quality control per

Question

2. Suppose that a database contains ten thousand errors. Two quality control per- sonnel, Xavier and Yvonne, operate independently of each-other, and uncover an error in each item with probability 0.3 and 0.31, respectively, independently of all other errors uncovered. Denote by X the number of errors Xavier uncovers, and the number of errors Yvonne uncovers by Y (a) Appealing to the central limit theorem, what are the approximate Normal distributions of X and Y? (b) Using part (a) and the table of the standard Normal cdf, approximate the probability that Yvonne uncovers more errors than Xavier

Explanation / Answer

a) here mean for Xavier =np=10000*0.3 =3000

and variance =np(1-p) =2100

hence approximate distribution for Xavier =N(3000,2100)

mean for Yvonne =np=10000*0.31 =3100

and variance =np(1-p) =2139

hence approximate distribution for Yvonne =N(3100,2139)

b)

here mean differnece between Yvonne and Xavier =3100-3000 =100

and std difference =(2100+2139)1/2 =65.1076

therefore probability that differnece in greater then 0 =P(X>0) =P(Z>(0-100)/65.1076)=P(Z>-1.5359)

=0.9377

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