How many work cycles should be timed to estimate the average cycle time to within 1 percent of the sample mean with a confidence of 95.0 percent if a pilot study yielded these times (in minutes): 4.9, 4.1, 4.3, 4.1, 4.3, and 4.6? (Round the final answer to the nearest whole number.)
Question
How many work cycles should be timed to estimate the average cycle time to within 1 percent of the sample mean with a confidence of 95.0 percent if a pilot study yielded these times (in minutes): 4.9, 4.1, 4.3, 4.1, 4.3, and 4.6? (Round the final answer to the nearest whole number.)
Solution
To answer this question, we need to follow these steps:
- Calculate the sample mean (average) of the pilot study times.
- Calculate the standard deviation of the pilot study times.
- Use the formula for estimating the sample size needed in a study, which is n = (Z^2 * σ^2) / E^2, where Z is the Z-score corresponding to the desired confidence level (1.96 for 95% confidence), σ is the standard deviation, and E is the desired margin of error (1% of the sample mean in this case).
Let's go through these steps:
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Calculate the sample mean: (4.9 + 4.1 + 4.3 + 4.1 + 4.3 + 4.6) / 6 = 4.38 minutes
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Calculate the standard deviation: First, find the variance, which is the average of the squared differences from the mean. Variance = [(4.9-4.38)^2 + (4.1-4.38)^2 + (4.3-4.38)^2 + (4.1-4.38)^2 + (4.3-4.38)^2 + (4.6-4.38)^2] / 6 = 0.046 Then, take the square root of the variance to get the standard deviation. Standard deviation = sqrt(0.046) = 0.2144
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Use the formula to estimate the sample size: n = (1.96^2 * 0.2144^2) / (0.01 * 4.38)^2 = 24.6
Since we can't have a fraction of a work cycle, we round this up to the nearest whole number, which is 25. So, 25 work cycles should be timed to estimate the average cycle time to within 1 percent of the sample mean with a confidence of 95.0 percent.
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