This question is incomplete, the complete question is;
A given population proportion is .25. What is the probability of getting each of the following sample proportions
a) n = 110 and = p^ ≤ 0.21, prob = ?
b) n = 33 and p^ > 0.24, prob = ?
Round all z values to 2 decimal places. Round all intermediate calculation and answers to 4 decimal places.)
Answer:
a) the probability of getting the sample proportion is 0.1660
b) the probability of getting the sample proportion is 0.5517
Explanation:
Given the data in the questions
a)
population proportion = 0.25
q = 1 - p = 1 - 0.25 = 0.75
sample size n = 110
mean = μ = 0.25
S.D = √( p( 1 - p) / n ) = √(0.25( 1 - 0.25) / 110 ) √( 0.1875 / 110 ) = 0.0413
Now, P( p^ ≤ 0.21 )
= P[ (( p^ - μ ) /S.D) < (( 0.21 - μ ) / S.D)
= P[ Z < ( 0.21 - 0.25 ) / 0.0413)
= P[ Z < -0.04 / 0.0413]
= P[ Z < -0.97 ]
from z-score table
P( X ≤ 0.21 ) = 0.1660
Therefore, the probability of getting the sample proportion is 0.1660
b)
population proportion = 0.25
q = 1 - p = 1 - 0.25 = 0.75
sample size n = 33
mean = μ = 0.25
S.D = √( p( 1 - p) / n ) = √(0.25( 1 - 0.25) / 33 ) = √( 0.1875 / 33 ) = 0.0754
Now, P( p^ > 0.24 )
= P[ (( p^ - μ ) /S.D) > (( 0.24 - μ ) / S.D)
= P[ Z > ( 0.24 - 0.25 ) / 0.0754 )
= P[ Z > -0.01 / 0.0754 ]
= P[ Z > -0.13 ]
= 1 - P[ Z < -0.13 ]
from z-score table
{P[ Z < -0.13 ] = 0.4483}
1 - 0.4483
P( p^ > 0.24 ) = 0.5517
Therefore, the probability of getting the sample proportion is 0.5517