Question

Generation Y has been defined as those individuals who were born between 1981 abd 1991. A 2010 survry by a credit counseling foundation found that 53% of the young adults in Generation Y pay their monthly bills on time. Suppose we take a random sample of 180 people from generation Y. Complete parts A through E below.

a. calculate standard error of the proportion

b. what is the probability that 120 or fewer will pay monthly bills on time

c. what is the probability that 95 or fewer will pay monthly bills on time

d.what is the probability that 111 or more will pay monthly bills on time

e. probbaility that between 97 and 112 will pay on time

Answer 1

Mean = n * P = ( 180 * 0.53 ) = 95.4
Variance = n * P * Q = ( 180 * 0.53 * 0.47 ) = 44.838
Standard deviation = = 6.6961

Part a)

Standard error of the proportion =

Part b)

P ( X <= 120 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 120 + 0.5 ) = P ( X < 120.5 )

P ( X < 120.5 )
Standardizing the value

Z = ( 120.5 - 95.4 ) / 6.6961
Z = 3.75

P ( X < 120.5 ) = P ( Z < 3.75 )
P ( X < 120.5 ) = 0.9999

Part c)

P ( X <= 95 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 95 + 0.5 ) = P ( X < 95.5 )

P ( X < 95.5 )
Standardizing the value

Z = ( 95.5 - 95.4 ) / 6.6961
Z = 0.01

P ( X < 95.5 ) = P ( Z < 0.01 )
P ( X < 95.5 ) = 0.504

Part d)

P ( X >= 111 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 111 - 0.5 ) =P ( X > 110.5 )

P ( X > 110.5 ) = 1 - P ( X < 110.5 )
Standardizing the value

Z = ( 110.5 - 95.4 ) / 6.6961
Z = 2.26

P ( Z > 2.26 )
P ( X > 110.5 ) = 1 - P ( Z < 2.26 )
P ( X > 110.5 ) = 1 - 0.9881
P ( X > 110.5 ) = 0.0119

Part e)

P ( 97 < X < 112 )
Using continuity correction
P ( n + 0.5 < X < n - 0.5 ) = P ( 97 + 0.5 < X < 112 - 0.5 ) = P ( 97.5 < X < 111.5 )

P ( 97.5 < X < 111.5 )
Standardizing the value

Z = ( 97.5 - 95.4 ) / 6.6961
Z = 0.31
Z = ( 111.5 - 95.4 ) / 6.6961
Z = 2.4
P ( 0.31 < Z < 2.4 )
P ( 97.5 < X < 111.5 ) = P ( Z < 2.4 ) - P ( Z < 0.31 )
P ( 97.5 < X < 111.5 ) = 0.9919 - 0.6231
P ( 97.5 < X < 111.5 ) = 0.3688