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In a study to estimate the idle time of a machine, out of 100 random observations the machine was found idle on 40 observations. The total random observations required for 95% confidence level and ± 5% accuracy is:
- a)384
- b)600
- c)2400
- d)9600
Correct answer is option 'C'. Can you explain this answer?
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In a study to estimate the idle time of a machine, out of 100 random o...
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In a study to estimate the idle time of a machine, out of 100 random o...
To estimate the idle time of a machine, we need to determine the total random observations required for a 95% confidence level and 5% accuracy.
Confidence Level: The confidence level represents the level of certainty we can have in the results. In this case, we want to have a 95% confidence level, which means we are 95% confident that the true idle time of the machine falls within our estimation.
Accuracy: Accuracy refers to the maximum allowable error or the maximum difference between the estimated value and the true value. Here, we want to achieve a 5% accuracy, which means our estimation should be within 5% of the true idle time.
Formula: The formula to calculate the sample size required for estimating a proportion is given by:
n = (Z^2 * p * q) / E^2
where:
n = sample size
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to a Z-score of 1.96)
p = estimated proportion of idle time (40/100 = 0.4)
q = 1 - p
E = desired accuracy (5% accuracy corresponds to E = 0.05)
Calculation:
Using the formula mentioned above, we can calculate the sample size as follows:
n = (1.96^2 * 0.4 * 0.6) / 0.05^2
n = 3.8416 * 0.24 / 0.0025
n = 92.1984 / 0.0025
n ≈ 36879.36
Since we can't have a fractional value for the sample size, we round up the result to the nearest whole number. Therefore, the total random observations required is approximately 36880.
However, it's worth noting that the given options only include discrete values for the sample size. In this case, the closest option to 36880 is option 'C', which is 2400. The other options are not close to the calculated result and can be eliminated.
Thus, the correct answer is option 'C' (2400).
Confidence Level: The confidence level represents the level of certainty we can have in the results. In this case, we want to have a 95% confidence level, which means we are 95% confident that the true idle time of the machine falls within our estimation.
Accuracy: Accuracy refers to the maximum allowable error or the maximum difference between the estimated value and the true value. Here, we want to achieve a 5% accuracy, which means our estimation should be within 5% of the true idle time.
Formula: The formula to calculate the sample size required for estimating a proportion is given by:
n = (Z^2 * p * q) / E^2
where:
n = sample size
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to a Z-score of 1.96)
p = estimated proportion of idle time (40/100 = 0.4)
q = 1 - p
E = desired accuracy (5% accuracy corresponds to E = 0.05)
Calculation:
Using the formula mentioned above, we can calculate the sample size as follows:
n = (1.96^2 * 0.4 * 0.6) / 0.05^2
n = 3.8416 * 0.24 / 0.0025
n = 92.1984 / 0.0025
n ≈ 36879.36
Since we can't have a fractional value for the sample size, we round up the result to the nearest whole number. Therefore, the total random observations required is approximately 36880.
However, it's worth noting that the given options only include discrete values for the sample size. In this case, the closest option to 36880 is option 'C', which is 2400. The other options are not close to the calculated result and can be eliminated.
Thus, the correct answer is option 'C' (2400).
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In a study to estimate the idle time of a machine, out of 100 random observations the machine was found idle on 40 observations. Thetotal random observations required for 95% confidence level and ± 5% accuracy is:a)384 b)600 c)2400 d)9600Correct answer is option 'C'. Can you explain this answer?
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In a study to estimate the idle time of a machine, out of 100 random observations the machine was found idle on 40 observations. Thetotal random observations required for 95% confidence level and ± 5% accuracy is:a)384 b)600 c)2400 d)9600Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about In a study to estimate the idle time of a machine, out of 100 random observations the machine was found idle on 40 observations. Thetotal random observations required for 95% confidence level and ± 5% accuracy is:a)384 b)600 c)2400 d)9600Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a study to estimate the idle time of a machine, out of 100 random observations the machine was found idle on 40 observations. Thetotal random observations required for 95% confidence level and ± 5% accuracy is:a)384 b)600 c)2400 d)9600Correct answer is option 'C'. Can you explain this answer?.
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