Question

The opera theater manager calculates that 15% of the opera tickets for tonight's show have been sold. If the manager is accurate, what is the probability that the proportion of tickets sold in a sample of 690 tickets would differ from the population proportion by more than 4% ? Round your answer to four decimal places.

Answer 1

Solution:

Given ,

p = 15% = 0.15 (population proportion)

1 - p = 0.85

n = 690  (sample size)

Let be the sample proportion.

The sampling distribution of is approximately normal with

mean = =  p = 0.15

SD =   =     

=    

=  0.01359347669

Now ,

P( will differ from p by more than 4%)

=  P( will differ from p by more than 0.04)

= 1 - P( will differ from μ by less than 0.04)

= 1 - P(p - 0.04  <   < p + 0.04 )

= 1 - P( 0.15 - 0.04 < < 0.15 + 0.04)

= 1 - P(0.11 < < 0.19)

= 1 - { P( < 0.19) - P( < 0.11) }

= 1 - { P(Z <(0.19 - 0.15)/0.01359347669) - P(Z <(0.11 - 0.15)/0.01359347669) }

= 1 - { P(Z < 2.94) - P(Z < -2.94) }

= 1 - { 0.9984 - 0.0016} .. (use z table)

= 1 - 0.9968

= 0.0032