Question

In a​ lottery, you bet on a six-digit number between 000000 and 111111. For a​ $1 bet, you win ​$700,000 if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are μ=0.70 (that is, 70 cents) and σ=700.00. Joan figures that if she plays enough times every​ day, eventually she will strike it​ rich, by the law of large numbers. Over the course of several​ years, she plays 1 million times. Let x denote her average winnings.

a. Find the mean and standard deviation of the sampling distribution of x

b. About how likely is it that Joan​'s average winnings exceed​ $1, the amount she paid to play each​ time? Use the central limit theorem to find an approximate answer.

Answer 1

P(z<Z) table:

a.

for sampling distribution:

mean = population mean = 0.70 dollars

SD = SDpopulation / (n^0.5) = 700/(1,000,000^0.5)

SD = 0.7

b.

z = (sample mean - pop. mean)/SD

for z = $1

z = (1-0.70)/0.7

= 0.3/0.7

= 0.43

P(smaple mean < $1) = P(z<0.43)

= 0.6664 {from table}

P(sample mean > $1) = 1 - P(smaple mean < $1)

= 1 - 0.6664

= 0.3336

(please upvote)