Question

A global research study found that the majority of today's working women would prefer a better work-life balance to an increased salary. One of the most important contributors to work-life balance identified by the survey was "flexibility," with 41% of women saying that having a flexible work schedule is either very important or extremely important to their career success. Suppose you select a sample of 100 working women.

The probability that in the sample fewer than 48​% say that having a flexible work schedule is either very important or extremely important to their career success is

The probability that in the sample between 33​% and 48​% say that having a flexible work schedule is either very important or extremely important to their career success is

The probability that in the sample more than 42% say that having a flexible work schedule is either very important or extremely important to their career success is

Answer 1

Sampling distribution of p̂ is approximately normal if np >=10 and n (1-p) >= 10
n * p = 100 * 0.41 = 41
n * (1 - p ) = 100 * (1 - 0.41) = 59

Mean = = p = 0.41
Standard deviation = = 0.049183

Part a)

X ~ N ( µ = 0.41 , σ = 0.049183 )
P ( X < 0.48 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.48 - 0.41 ) / 0.049183
Z = 1.4233
P ( ( X - µ ) / σ ) < ( 0.48 - 0.41 ) / 0.049183 )
P ( X < 0.48 ) = P ( Z < 1.4233 )
P ( X < 0.48 ) = 0.9227

Part b)

X ~ N ( µ = 0.41 , σ = 0.049183 )
P ( 0.33 < X < 0.48 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.33 - 0.41 ) / 0.049183
Z = -1.6266
Z = ( 0.48 - 0.41 ) / 0.049183
Z = 1.4233
P ( -1.63 < Z < 1.42 )
P ( 0.33 < X < 0.48 ) = P ( Z < 1.42 ) - P ( Z < -1.63 )
P ( 0.33 < X < 0.48 ) = 0.9227 - 0.0519
P ( 0.33 < X < 0.48 ) = 0.8708

Part c)

X ~ N ( µ = 0.41 , σ = 0.049183 )
P ( X > 0.42 ) = 1 - P ( X < 0.42 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.42 - 0.41 ) / 0.049183
Z = 0.2033
P ( ( X - µ ) / σ ) > ( 0.42 - 0.41 ) / 0.049183 )
P ( Z > 0.2033 )
P ( X > 0.42 ) = 1 - P ( Z < 0.2033 )
P ( X > 0.42 ) = 1 - 0.5805
P ( X > 0.42 ) = 0.4195