Question

in the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2, 36. 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 $35, otherwise you lose $1. complete parts (a) through (g) below click here to view the standard normal distribution table (page 1) click here to view the standard normal distribution table (page 2) (c) suppose that you play the game 90 times so that ( n = 90 ). describe the sampling distribution of ( overline{x} ), the mean amount won per game. the sample mean ( overline{x} ) is approximately normal what are the mean and standard deviation of the sampling distribution of ( overline{x} )? round your results to the nearest penny ( mu_{overline{x}}=-0.05 ) ( sigma_{overline{x}} = 0.61 ) (d) what is the probability of being ahead after playing the game 90 times? that is, what is the probability that the sample mean is greater than 0 for ( n = 90 )? ( p(overline{x}>0)= ) (type an integer or decimal rounded to four decimal places as needed.)

Explanation:

Step1: Standardize the value

Use the formula \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}\). Here, \(\bar{x} = 0\), \(\mu_{\bar{x}}=- 0.05\), and \(\sigma_{\bar{x}}=0.61\).

$$z=\frac{0 - (-0.05)}{0.61}=\frac{0.05}{0.61}\approx0.08$$

Step2: Find the probability

We want \(P(\bar{X}>0)\), which is equivalent to \(P(Z > 0.08)\). Since \(P(Z>z)=1 - P(Z\leq z)\), and from the standard - normal distribution table, \(P(Z\leq0.08) = 0.5319\).

$$P(\bar{X}>0)=1 - P(Z\leq0.08)=1 - 0.5319 = 0.4681$$

Answer:

\(0.4681\)