How many work cycles should be timed to estimate the average cycle time to within 2 percent of the sample mean with a confidence of 99.0 percent if a pilot study yielded these times (minutes): 5.2, 5.5, 6.6, 5.3, 5.5, and 5.1? The standard deviation is 0.547 minutes per cycle. (Use the "mean time" value to 2 decimal places and other values to 3 decimal places for intermediate calculations. Round up your final answer to the next whole number.)

Question

How many work cycles should be timed to estimate the average cycle time to within 2 percent of the sample mean with a confidence of 99.0 percent if a pilot study yielded these times (minutes): 5.2, 5.5, 6.6, 5.3, 5.5, and 5.1? The standard deviation is 0.547 minutes per cycle. (Use the "mean time" value to 2 decimal places and other values to 3 decimal places for intermediate calculations. Round up your final answer to the next whole number.)

Answer 1

To estimate the average cycle time within 2 percent of the sample mean with a confidence of 99 percent, we need to determine the number of work cycles to be timed.

Here's the step-by-step process to calculate it:

1. Calculate the sample mean (x̄) of the given data:
x̄ = (5.2 + 5.5 + 6.6 + 5.3 + 5.5 + 5.1) / 6 = 5.45 minutes (rounded to 2 decimal places)

2. Determine the critical z-value for the desired confidence level of 99 percent.

The confidence level is the complement of the significance level (α), often expressed as (1 - α). In this case, the significance level is 1 - 0.99 = 0.01.

To find the z-value associated with a 0.01 significance level, we consult the standard normal distribution table or use statistical software. The z-value for a 0.01 significance level is approximately 2.576 (rounded to 3 decimal places).

3. Calculate the margin of error (MoE):
MoE = (z-value * standard deviation) / √(n)

In this case, the standard deviation is given as 0.547 minutes per cycle. We'll use the approximate value of the z-value, 2.576.

MoE = (2.576 * 0.547) / √(n)

4. Rearrange the equation to solve for n:
n = ((2.576 * 0.547) / MoE)^2

We want the result to be rounded up to the next whole number, so we use the ceiling function to round up.

n = ceiling(((2.576 * 0.547) / MoE)^2)

5. Substitute the values into the equation:
n = ceiling(((2.576 * 0.547) / (0.02 * 5.45))^2)
n = ceiling((1.40881252)^2)
n = ceiling(1.985003)
n = 2

Therefore, we should time at least 2 work cycles to estimate the average cycle time to within 2 percent of the sample mean with a confidence of 99 percent.

Note: It's important to note that the number of cycles needed for estimation may not always result in whole numbers, so rounding up to the next whole number is a practical approach in these cases.