Question

Approximately 60% of all part-time college students in the United States are female. In random samples of size 100 taken from the population of all part-time college students:

• Describe the sampling distribution of the sample proportion of part-time female college students.

• What is the probability that the sample proportion of part-time female college students is at most 0.51?

• What is the probability that the sample proportion of part-time female college students is at least 0.67?

• What is the probability that the sample proportion of part-time female college students is between 0.53 and 0.60?

Answer 1

Solution

Given that,

p = 0.60

1 - p = 1-0.60 = 0.40

n = 100

= p = 0.60

=  [p ( 1 - p ) / n] =   [0.60*(0.40) /100 ] = 0.0490

a) P( ≤ 0.51 )

= P[( - ) / ≤ (0.51 - 0.60) /0.0490 ]

= P(z ≤ -1.84)

probability = 0.0329

b)

P( ≥ 0.67 ) = 1 - P( ≤ 0.67)

= 1 - P(( - ) / ≤ ( 0.67 -0.60) /0.0490 )

= 1 - P(z ≤ 1.43 )

= 1 - 0.9236 = 0.0764

probability = 0.0764

c)

P( 0.53< < 0.60)  

= P[(0.53 - 0.60) /0.0490 < ( - ) / < (0.60-0.60) /0.0490 ]

= P( -1.43< z < 0 )

= P(z < 0) - P(z < -1.43 )

= 0.5 - 0.0764 = 0.4236

probability = 0.4236