In the game of roulette, a wheel consists of 42 slots numbered 0, 00, 1, 2,...,

Question

In the game of roulette, a wheel consists of 42 slots numbered 0, 00, 1, 2,..., 40. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. If the number of the slot the ball falls into matches the number you selected, you win $40; otherwise you lose $1.

a.) Determine the mean and standard deviation of the random variable X.

b.) Suppose that you play the game 25 times so that . Describe the sampling distribution of , the mean amount won per game.

c.) Suppose that you play the game 64 times so that . Describe the sampling distribution of , the mean amount won per game.

d.) Suppose that you play the game 400 times so that . Describe the sampling distribution of , the mean amount won per game.

Explanation / Answer

here probability of winning $40 =P(X=40)=1/42

and probability of losing $1 =P(X=-1)=41/42

a) from above:

therfore mean of X =-0.024

and std deviation =6.2507

b) from Central limit theorum:

sampling distribution of mean will be approximately normal with :

sample mean =-0.024

and std error of mean =6.2507/(25)1/2 =1.25

c)

sampling distribution of mean will be approximately normal with :

sample mean =-0.024

and std error of mean =6.2507/(64)1/2 =0.7813

d)

sampling distribution of mean will be approximately normal with :

sample mean =-0.024

and std error of mean =6.2507/(400)1/2 =0.3125

x p(x) xP(x) x2P(X) (x-)2 (x-)2P(x) -1 41/42 -0.976 0.976 0.953 0.930 40 1/42 0.952 38.095 1601.905 38.141 total 1 = -0.024 39.071 1602.858 2= 39.0709 std deviation=     =    2 = 6.2507

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