Question

1.            The mean wait time at Social Security Offices is 25 minutes with a standard deviation of 11 minutes. Use this information to answer the following questions:

A.            If you randomly select 40 people what is the probability that their average wait time will be more than 27 minutes?

B.            If you randomly select 75 people what is the probability that their average wait time will be between 23 and 26 minutes?

C.            If you randomly select 100 people what is the probability that their average wait time will be at most 24 minutes?

(1 point each)

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

1)

a)

Here, μ = 25, σ = 1.7393 and x = 27. We need to compute P(X >= 27). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (27 - 25)/1.7393 = 1.15

Therefore,
P(X >= 27) = P(z <= (27 - 25)/1.7393)
= P(z >= 1.15)
= 1 - 0.8749 = 0.1251

b)

Here, μ = 25, σ = 1.2702, x1 = 23 and x2 = 26. We need to compute P(23<= X <= 26). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (23 - 25)/1.2702 = -1.57
z2 = (26 - 25)/1.2702 = 0.79

Therefore, we get
P(23 <= X <= 26) = P((26 - 25)/1.2702) <= z <= (26 - 25)/1.2702)
= P(-1.57 <= z <= 0.79) = P(z <= 0.79) - P(z <= -1.57)
= 0.7852 - 0.0582
= 0.7270

c)

Here, μ = 25, σ = 1.1 and x = 24. We need to compute P(X <= 24). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (24 - 25)/1.1 = -0.91

Therefore,
P(X <= 24) = P(z <= (24 - 25)/1.1)
= P(z <= -0.91)
= 0.1814

2)

a)

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