Question

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 45​% 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 53​% 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 41​% and 53​% 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 47​% say that having a flexible work schedule is either very important or extremely important to their career success is ___.

Answer 1

a)

Here, μ = 0.45, σ = 0.0497 and x = 0.53. We need to compute P(X <= 0.53). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0.53 - 0.45)/0.0497 = 1.61

Therefore,
P(X <= 0.53) = P(z <= (0.53 - 0.45)/0.0497)
= P(z <= 1.61)
= 0.9463

b)

Here, μ = 0.45, σ = 0.0497, x1 = 0.41 and x2 = 0.53. We need to compute P(0.41<= X <= 0.53). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.41 - 0.45)/0.0497 = -0.8
z2 = (0.53 - 0.45)/0.0497 = 1.61

Therefore, we get
P(0.41 <= X <= 0.53) = P((0.53 - 0.45)/0.0497) <= z <= (0.53 - 0.45)/0.0497)
= P(-0.8 <= z <= 1.61) = P(z <= 1.61) - P(z <= -0.8)
= 0.9463 - 0.2119
= 0.7344

c)

Here, μ = 0.45, σ = 0.0497 and x = 0.47. We need to compute P(X >= 0.47). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0.47 - 0.45)/0.0497 = 0.4

Therefore,
P(X >= 0.47) = P(z <= (0.47 - 0.45)/0.0497)
= P(z >= 0.4)
= 1 - 0.6554 = 0.3446