Question

Exercise 6-50

The accounting department at Weston Materials Inc., a national manufacturer of unattached garages, reports that it takes two construction workers a mean of 30 hours and a standard deviation of 5 hours to erect the Red Barn model. Assume the assembly times follow the normal distribution. Refer to the table in Appendix B.1.

a-1. Determine the z-values for 26 and 35 hours. (Negative answers should be indicated by a minus sign. Round the final answers to 1 decimal place.)

26 hours corresponds to z =           

35 hours corresponds to z =           

a-2. What percentage of the garages take between 30 hours and 35 hours to erect? (Round the final answer to 2 decimal places.)

Percentage            %

b. What percentage of the garages take between 26 hours and 35 hours to erect? (Round the final answer to 2 decimal places.)

Percentage            %

c. What percentage of the garages take 25.4 hours or less to erect? (Round the final answer to 2 decimal places.)

Percentage            %

d. Of the garages, 10% take how many hours or more to erect? (Round the final answer to 1 decimal place.)

Hours           

Answer 1

a1)

Here, μ = 30, σ = 5, x1 = 26 and x2 = 35. We need to compute P(26<= X <= 35). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (26 - 30)/5 = -0.8
z2 = (35 - 30)/5 = 1

26 hours corresponds to z = -0.8

35 hours corresponds to z = 1

a2)

Here, μ = 30, σ = 5, x1 = 30 and x2 = 35. We need to compute P(30<= X <= 35). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (30 - 30)/5 = 0
z2 = (35 - 30)/5 = 1

Therefore, we get
P(30 <= X <= 35) = P((35 - 30)/5) <= z <= (35 - 30)/5)
= P(0 <= z <= 1) = P(z <= 1) - P(z <= 0)
= 0.8413 - 0.5
= 0.3413 = 34.13%

b)

Here, μ = 30, σ = 5, x1 = 26 and x2 = 35. We need to compute P(26<= X <= 35). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (26 - 30)/5 = -0.8
z2 = (35 - 30)/5 = 1

Therefore, we get
P(26 <= X <= 35) = P((35 - 30)/5) <= z <= (35 - 30)/5)
= P(-0.8 <= z <= 1) = P(z <= 1) - P(z <= -0.8)
= 0.8413 - 0.2119
= 0.6294 = 62.94%

c)

Here, μ = 30, σ = 5 and x = 25.4. We need to compute P(X <= 25.4). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (25.4 - 30)/5 = -0.92

Therefore,
P(X <= 25.4) = P(z <= (25.4 - 30)/5)
= P(z <= -0.92)
= 0.1788

= 17.88%

d)

z value at top 10% = 1.28

z = (x - mean)/s
1.28 = (x - 30)/5
x = 5 *1.28 + 30
x = 36.4