Question

A random sample of size n = 130 is taken from a population with a population proportion p = 0.58. (You may find it useful to reference the z table.)

a. Calculate the expected value and the standard error for the sampling distribution of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.)

b. What is the probability that the sample proportion is between 0.50 and 0.70? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. What is the probability that the sample proportion is less than 0.50? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

Answer 1

Solution:

Given that,

n = 130

= 0.58

1 - = 1 - 0.58 = 093

a )   =   = 0.542

= ( 1 - ) / n

=   0.58* 0.42 / 130

= 0.0433

= 0.0433

P( 0.50 < < 0.70 )  

b )P( 0.50 - 0.58 / 0.0433 ) < ( -    / ) < ( 0.70 - 0.58 / 0.0433 )

P ( - 0.08 /0.0433< z < 0.12 / 0.0433)

P ( - 1.85 < z < 2.77 )

P ( z < 2.77 ) - p ( z < -1.85 )

Using z table

= 0.9972 - 0.0322

= 0.9650

Probability = 0.9650

c ) P (   < 0.50 )

P ( -    / ) < ( 0.50 - 0.58 / 0.0433 )

P ( z < - 0.08 / 0.0433 )

Using z table

P ( z < -1.85 )

=0.0322

Probability = 0.0322