Question

2.            The proportion of people who wait more than an hour at the Social Security Office is 28%. Use this information to answer the following questions:

A.            If you randomly select 45 people what is the probability that at least 34% of them will wait more than an hour?

B.            If you randomly select 60 people what is the probability that between 25% and 30% of them will wait more than an hour?

C.            If you randomly select 150 people what is the probability that less than 23% of them will wait more than an hour?

Answer 1

a)

Here, μ = 0.28, σ = 0.0669 and x = 0.34. We need to compute P(X >= 0.34). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0.34 - 0.28)/0.0669 = 0.9

Therefore,
P(X >= 0.34) = P(z <= (0.34 - 0.28)/0.0669)
= P(z >= 0.9)
= 1 - 0.8159 = 0.1841

b)

Here, μ = 0.28, σ = 0.0578, x1 = 0.25 and x2 = 0.3. We need to compute P(0.25<= X <= 0.3). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.25 - 0.28)/0.0578 = -0.52
z2 = (0.3 - 0.28)/0.0578 = 0.34

Therefore, we get
P(0.25 <= X <= 0.3) = P((0.3 - 0.28)/0.058) <= z <= (0.3 - 0.28)/0.058)
= P(-0.52 <= z <= 0.34) = P(z <= 0.34) - P(z <= -0.52)
= 0.6368 - 0.3015
= 0.3353

c)

Here, μ = 0.28, σ = 0.0367 and x = 0.23. We need to compute P(X <= 0.23). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0.23 - 0.28)/0.0367 = -1.36

Therefore,
P(X <= 0.23) = P(z <= (0.23 - 0.28)/0.0367)
= P(z <= -1.36)
= 0.0869