Question

Reconsider the game of roulette. Recall that when you bet $1 on a color, you have an 18/38 probability of winning $1 and a 20/38 probability of losing $1 (for a net winnings of -$1). Consider playing for a random sample of n = 4 spins, and consider the statistic x-bar = sample mean of your net winnings per spin.
a) Determine the (exact) sampling distribution of x-bar. [Hint: Start by listing the possible values of x-bar. Then use the binomial distribution to help with calculating the probability of each possible value.]
b) Use your answer to a) to determine the probability that x-bar > 0. Now suppose that you play roulette, again betting $1 on a color each time, for a random sample of n = 100 spins. Again consider the statistic x-bar = sample mean of your net winnings per spin.

c) Describe what the Central Limit Theorem says about the (approximate) sampling distribution of x-bar.

d) Use your answer to c) and the normal distribution to determine the (approximate) probability that x-bar > 0.

e) How would this probability (that x-bar > 0) change as the sample size increases? Explain/justify your answer.

Answer 1