Government data show that 9 of the American population are a

Government data show that 9% of the American population are at least 75 years of age. To test a random-digit dialing device, you use the device to call randomly chosen residential telephones in your county. Of the 250 members of the households contacted, 4% are 75 years or older.

What is the probability that a random sample of 250 respondents would contain less than 4% of individuals at least 75 years of age? (Use 3 decimal places)

Solution

Government data show that 9% of the American population are at least 75 years of age. To test a random-digit dialing device, you use the device to call randomly chosen residential telephones in your county. Of the 250 members of the households contacted, 4% are 75 years or older.

What is the probability that a random sample of 250 respondents would contain less than 4% of individuals at least 75 years of age? (Use 3 decimal places)

n=250

p=0.09

Expectation = np = 22.5

Variance = np(1 - p) = 20.475

Standard deviation = 4.5249

4% of 250 =10

Normal approximation to binomial used

For < 4%, With correction, 10 becomes 9.5

Z value for 9.5, z=(9.5-22.5)/4.5249 = -2.87

P( x<10) = ( z <-2.87) =0.0021

The required P=0.002   ( 3 decimals)