Concept explainers
The manufacture of a certain part requires two different machine operations. The time on machine 1 has
a. What is the
b. What is the probability that the total time used by machine 2 is less than 55 hours?
c. What is the probability that the total time used by both machines together is greater than 115 hours?
d. What is the probability that the total time used by machine 1 is greater than the total time used by machine 2?
a.
Find the probability that the total time used by machine 1 is greater than 55 hours.
Answer to Problem 18E
The probability that the total time used by machine 1 is greater than 55 hours is 0.1056.
Explanation of Solution
Given info:
The mean and standard deviation of the time on machine 1 is 0.5 hours and 0.4 hours. The mean and standard deviation of the time on machine 2 is 0.6 hours and 0.5 hours. The time taken by both the machines are independent. The total number of parts manufactured is 100.
Calculation:
The Central Limit Theorem:
The random variables
The sample mean is
Then if n is sufficiently large,
The random variables
Then for machine 1, the mean is,
The total time on machine 1 is
Then by the central limit theorem
Mean:
Substitute n as 100 and
Standard deviation:
Substitute n as 100 and
Also by the central limit theorem
Mean:
Substitute n as 100 and
Standard deviation:
Substitute n as 100 and
The required probability is,
The formula to convert
Substitute 50 for
The above probability can be obtained by finding the areas to the left of 1.25.
The shaded region represents the area to the right of 1.25 is shown below:
Use Table A.2: Cumulative Normal Distribution to find the area.
Procedure:
For z at 1.25,
- Locate 1.2 in the left column of the Table A.2.
- Obtain the value in the corresponding row below 0.05.
That is,
Then,
Thus, probability that the total time used by machine 1 is greater than 55 hours is 0.1056.
b.
Find the probability that the total time used by machine 2 is less than 55 hours.
Answer to Problem 18E
The probability that the total time used by machine 2 is less than 55 hours is 0.1587.
Explanation of Solution
Calculation:
The required probability is,
Substitute 60 for
The above probability can be obtained by finding the areas to the left of –1.
The shaded region represents the area to the left of –1 is shown below:
Use Table A.2: Cumulative Normal Distribution to find the area.
Procedure:
For z at –1,
- Locate –1.0 in the left column of the Table A.2.
- Obtain the value in the corresponding row below 0.00.
That is,
Then,
Thus, the probability that the total time used by machine 2 is less than 55 hours is 0.1587.
c.
Find the probability that total time used by both machines together is greater than 115 hours.
Answer to Problem 18E
The probability that total time used by both machines together is greater than 115 hours is 0.2177.
Explanation of Solution
Calculation:
Result:
Assume that X and Y are independent random variables with X follows Normal with mean
Total time used by both machines is denoted as
Substitute
The standard deviation is,
Substitute
The required probability is,
Substitute 110 for
The above probability can be obtained by finding the areas to the left of 0.78.
The shaded region represents the area to the right of 0.78 is shown below:
Use Table A.2: Cumulative Normal Distribution to find the area.
Procedure:
For z at 0.78,
- Locate 0.7 in the left column of the Table A.2.
- Obtain the value in the corresponding row below 0.08.
That is,
Then,
Thus, the value of
d.
Find the probability that total time used by both machine 1 is greater than the total time used by machine 2.
Answer to Problem 18E
The probability that total time used by both machine 1 is greater than the total time used by machine 2 is 0.0594.
Explanation of Solution
Calculation:
The difference between the time on machine 1 and the time on machine2 is denoted as
Substitute
The standard deviation is,
Substitute
The required probability is,
Substitute –10 for
The above probability can be obtained by finding the areas to the left of 1.56.
The shaded region represents the area to the right of 1.56 is shown below:
Use Table A.2: Cumulative Normal Distribution to find the area.
Procedure:
For z at 1.56,
- Locate 1.5 in the left column of the Table A.2.
- Obtain the value in the corresponding row below 0.06.
That is,
Then,
Thus, the value of
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Chapter 4 Solutions
Statistics for Engineers and Scientists
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,