Question

A hotel manager calculates that 33% of the hotel rooms are booked. If the manager is correct, what is the probability that the proportion of rooms booked in a sample of 405 rooms would differ from the population proportion by greater than 3% ? Round your answer to four decimal places.

Answer 1

Solution

Given that,

p = 33% = 0.33

1 - p = 1 - 0.33 = 0.67

n = 405

= p = 0.33

=  [p ( 1 - p ) / n] = [(0.33 * 0.67) / 405 ] = 0.0234

0.33 - 0.03 = 0.30 and 0.33 + 0.03 = 0.36

P( 0.30 < < 0.36 )

= P[( 0.30 - 0.33 ) / 0.0234 < ( - ) / < ( 0.36 - 0.33 ) / 0.0234 ]

= P( - 1.28 < z < 1.28 )

= P(z < 1.28 ) - P(z < -1.28 )

Using z table

= 0.8997 - 0.1003

= 0.7994