3) In May 1983,after an extensive investigation by the Consumer Product Safety C

Question

3) In May 1983,after an extensive investigation by the Consumer Product Safety Commission, Honeywell agreed to recall 770,000 potentially defective smoke detectors. The Commission suggested that about 40% of the Honeywell detectors were defective. However, Honeywell found only four defective sample of 2000 detectors and claimed that the recall was not justified. What is the probability of finding at most 4 defective detectors in a random sample of 2000 if, in fact, 40% of all detectors are defective? 4) Consider the following game: A cup is filled with 100 pennies. The cup is shaken, and the pennies are poured onto a table. If at least 60 of the pennies are Heads, you win $20. Otherwise, you lose $1. Use the normal approximation to the binomial, together with expected value, to decide if this is a good game to play

Explanation / Answer

3) a) here mean number of defective in 2000 detectors =np=2000*0.4=800

std deviation =(np(1-p))1/2 =21.9089

z score corresponding to 4 defective =(X-mean)/std deviaiton =(4-800)/21.9089 =-36.3323

as z score is very low probbaility of such an incident is extremely rare( ~ 0) if 40% defective are there

4) mean number of heads =np=100*0.5 =50

std deviation =(np(1-p))1/2 = 5

hence probabilty of losing =P(X<=59)=P(Z<(59.5-50)/5)=P(Z<1.9)=0.9713

and probability of winning =1-0.9713 =0.0287

hence expected value =expected gain -expected lose =0.0287*20 -0.9713*1=-$30.970

as expected value is negative it is not a good game to play.

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