Question

The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 50 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.1 pounds, what is the probability that the sample mean will be each of the following?

Appendix A Statistical Tables

a. More than 58 pounds
b. More than 57 pounds
c. Between 56 and 58 pounds
d. Less than 55 pounds
e. Less than 47 pounds

Answer 1

a)

Here, μ = 56.8, σ = 1.7112 and x = 58. We need to compute P(X >= 58). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (58 - 56.8)/1.7112 = 0.7

Therefore,
P(X >= 58) = P(z <= (58 - 56.8)/1.7112)
= P(z >= 0.7)
= 1 - 0.758 = 0.2420

b)

Here, μ = 56.8, σ = 1.7112 and x = 57. We need to compute P(X >= 57). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (57 - 56.8)/1.7112 = 0.12

Therefore,
P(X >= 57) = P(z <= (57 - 56.8)/1.7112)
= P(z >= 0.12)
= 1 - 0.5478 = 0.4522

c)

Here, μ = 56.8, σ = 1.7112, x1 = 56 and x2 = 58. We need to compute P(56<= X <= 58). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (56 - 56.8)/1.7112 = -0.47
z2 = (58 - 56.8)/1.7112 = 0.7

Therefore, we get
P(56 <= X <= 58) = P((58 - 56.8)/1.7112) <= z <= (58 - 56.8)/1.7112)
= P(-0.47 <= z <= 0.7) = P(z <= 0.7) - P(z <= -0.47)
= 0.758 - 0.3192
= 0.4388

d)
Here, μ = 56.8, σ = 1.7112 and x = 55. We need to compute P(X <= 55). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (55 - 56.8)/1.7112 = -1.05

Therefore,
P(X <= 55) = P(z <= (55 - 56.8)/1.7112)
= P(z <= -1.05)
= 0.1469

e)
Here, μ = 56.8, σ = 1.7112 and x = 47. We need to compute P(X <= 47). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (47 - 56.8)/1.7112 = -5.73

Therefore,
P(X <= 47) = P(z <= (47 - 56.8)/1.7112)
= P(z <= -5.73)
= 0