n on the sampling distribution? 7.18 Playing roulette slots. A roulette wheel in

Question

n on the sampling distribution? 7.18 Playing roulette slots. A roulette wheel in Las Vega gas has 38 you bet a dollar on a particular number, you'll win e ball ends up in that slot and $0 otherwise. Roulette TRY If lly likely wheels are calibrated so that each outcome is equa a. Let a denote your winnings when you play once. State e probability distribution of X. (This also represents e population distribution you would get if you could play roulette an infinite number of times.) It has mean 0.921 and standard deviation 5.603 day b. You decide to play once a minute for 12 hours a for the next week, a total of 5040 times. Show that the sampling distribution of your sample mean has mean = 0.921 and standard deviation = 0.079. c. Refer to part b. Using the central limit theorem, find the probability that with this amount of roulette play ing, your mean winnings is at least $1, so that you have not lost money after this week of playing. (Hint: Find the probability that a normal random variable with mean 0.921 and standard deviation 0.079 exceeds 1.0.)

Explanation / Answer

a) below is pmf of X:

here probability of winning =P(X=35) =1/38

and probability of losing P(X=0) =37/38

b)as sample mean for 5040 games =(X1+X2+....Xn)/5040

hence

mean of sample means=(E(X1)+E(X2)+...+E(Xn))/5040 =(0.921+0.921+..0.921)/5040 =5040*0.92/5040

=0.921

standard deviation =((Var(X1)+Var(X2)+...+Var(Xn))/50402)1/2 =(5.603*5040/50402)1/2 =0.079

c)

hence probability of winning at least $1:

for normal distribution z score =(X-)/ here mean=       = 0.921 std deviation   == 5.6030 sample size       =n= 5040 std error=x=/n= 0.079

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