n a lottery, you bet on a sixsix-digit number between 000000000000 and 111111111

Question

n a lottery, you bet on a

sixsix-digit

number between

000000000000

and

111111111111.

For a $1 bet, you win

$700 comma 000700,000

if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are

muequals=0.70.7

(that is,

7070

cents) and

sigmaequals=700.00700.00.

JoanJoan

figures that if

sheshe

plays enough times every day, eventually

sheshe

will strike it rich, by the law of large numbers. Over the course of several years,

sheshe

plays 1 million times. Let

x overbarx

denote

herher

average winnings.

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

x overbarx.

b. About how likely is it that

JoanJoan's

average winnings exceed $1, the amount

sheshe

paid to play each time? Use the central limit theorem to find an approximate answer.

Explanation / Answer

a.

According to central limit theorem

Mean of sampling distribution = population mu = 0.7

Standard deviation of sampling distribution = sigma divided by root n

here n = 1 million = 1000,000

hence square root of n =1000

Standard deviation of sampling distribution = sigma divided by root n = 700/1000= 0.7

b

Probability of Joan average winning exceeding 1

z = (0.7-0.7 )/0.7 = 0

Using the normal distributin tables probability for the value of z = 0 is 0.5

Hence the probability = 0.5 or 50 percent

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