Question

fill the blank, and please explain why,thanks.

Check whether the binomial approximation is acceptable in each of the following hypergeometric situations. Check the accuracy of your approximation by using Excel to find the actual hypergeometric probability. (Round "n/N" answers to 2 decimal places and other answers to 4 decimal places.) a. N= 100, n= 4, s= 50, PIX = 3) b. N= 200, n= 13, s = 75, PIX = 3) c. N= 160, n= 13, s= 19, PIX= 2) d. N= 500, n= 6, 5= 320, P(X = 5) n N Binomial approximation acceptable Binomial approximation probability 0. 2499 Hypergeometric probability 0.2499 0.04 نه 0.07 unacceptable 0. 1356 ف 0.08 unacceptable 0.2869 ن 0.01 acceptable 0.2316 ن

Answer 1

Binomial approximation to Hypergeometric

If n is small compared with N. then the hypergeometric distribution can be approximated using the binomial distribution.

The rule of thumb is that this is valid if (nN) <= 0.05. In this case, we can use the binomial distribution with parameters n and p=s/N

NCA 100 Binomiol Approximation to Hypergeometric • If n is small compared with N, then the hypergeometric distribution can be approximated by using the binomial distribution , • The rule of thumb is that is valid if (/N) €0.05. In this case, we can we the binomiol distribution with parameters nand po Then W = mean = mpang 8?<versionce snea-n cts P(x=27= Cape (1-P) 0-2 = "x () C1-8) = 8 Hypergeometric pmf is PCX=n) - y for a N=100 = 4, 5=50 . - = 0.04 50.5 → Binomial approximation we acceptable PX = 3 = ( (- 59 13 = 0.25 0.2499 The command for finding POX-3] In excel :3 = BINOMDIST (3,4, 0.5, 0) (: Mese o for P(X=] & for PCXL ) By using My pergeomeloit ( x=30 - 5%, % 360 ex56 .299 3921225 Using Excel HYPGEOMDIST ( 3,4,50, 100) 6] N= zoo, p= 13 , S= 75, À -30 = 0.065 = 0.07 CRound up to 2digits) A 70.05 then Binomial approuinction is not acceptable By using appromiator by using Excel P(x=3 = =BINOS MOIST (3,13, 0.045,0) =0.0401. Hyper geometric disto by using Excel PCX= 3) = HYP G GOMDIST(3, 13, 75,200) = 0.1356 100% By using P(x=3] by os using appooximation and wing Hyper gemetric dist are not exact same

CI N=160, n=13, S=19 N = 1 = 0.0812 = 0.8 70.05 we cannot use Binomial appoonindia By using Binomial approximation probabilidy by wing Excel PCX=27 = BINONDIST (2,13, 0.11875, 0). = 0.6 273811869 = 0.2738 By using Hyper geometric probabwg by wing Excel px=2) - HYPGEOM DISTC 2,13, 19,160) 50: © 28 6849 = 0.2869 The probability wing binomial approximation ond using Hyperg some- -tric distribution are not some 0.0 0.G O OOOOOOOOOOOOOOOOOOOOOOOOOOO d] N-500, 7=6, 5=320 À - SOOS 0.012 = 0.01 Cround up to 2digits) Here A L0-05 then we we binomial approximation By using bim omia) approdi mation PCX= 6] = BINOMOIST (5,6, 0.64,0) 50.23 1928234 = 0.2319 By using. Nypeugeometric dist P= 5) < = H PREUMDIST(56 ,320,500) = 0,2316 2995 5 = 0.2316 The probability using binomial approxiation & Hypes geometric distribution cre exact same Binomial approximation Binomralappromotion Hypergesmelise probability probability aceptable 062499 0.2499 007 unodeptable 0.040) 0.1356 unacceptable 0.2738 0.2869 0.08 0.2316 0.2316 acceptable 0.01 0-04