According to a survey conducted by Harrahs Home Testing Inst

According to a survey conducted by Harrah\'s Home Testing Institute, adults from 50% of U.S households have gambled in a casino at some time in thier life (American Demographics, October 1993). Assume that this percentage is true for the current population of adults from all U.S households. A random sample of 400 U.S households is selected.

(a). Using normal approximation to find the probability that the indicated number of the 400 selecte household have gambled in a casino at some time in their life. (a). less than 200 (b). between 180 and 320 inclusive (c). at least 310

(b). Suppose a sample of 400 households shows that 320 have gambled in a casino at some time in their life. Is 320 an unusual number? Make your comments on this results.

Solution

Normal Approximation to Binomial Distribution
Mean ( np ) =400 * 0.5 = 200
Standard Deviation ( npq )= 400*0.5*0.5 = 10
Normal Distribution = Z= X- u / sd                   
a)
P(X < 200) = (200-200)/10
= 0/10= 0
= P ( Z <0) From Standard NOrmal Table
= 0.5                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 180) = (180-200)/10
= -20/10 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 320) = (320-200)/10
= 120/10 = 12
= P ( Z <12) From Standard Normal Table
= 1
P(180 < X < 320) = 1-0.02275 = 0.9772                  
c)
P(X >= 310) = (310-200)/10
= 110/10 = 11
= P ( Z >11) From Standard Normal Table
= 0