Given the following information, determine the sample size needed if the standard time estimate is to be within 5 percent of the true mean 99 percent of the time.
Work Element
Standard Deviation (minutes)
Mean Observed Time (minutes)
1
0.20
1.10
2
0.10
0.80
3
0.15
0.90
4
0.10
1.00
(Round your answer to the nearest whole number, the tolerance is +/-1.)
To determine the sample size needed, we can use the formula for sample size determination based on standard deviation and desired level of confidence.
The formula for calculating the required sample size is:
n = (Z * S / E)^2
Where:
n is the required sample size
Z represents the Z statistic for the desired level of confidence
S is the standard deviation
E is the desired margin of error (expressed as a proportion or percentage)
In this case, we want the standard time estimate to be within 5 percent of the true mean 99 percent of the time.
To start, let's calculate the value of Z for a 99 percent confidence level. Z is the number of standard deviations from the mean that corresponds to the desired confidence level. In this case, we need to use a one-tailed Z value since we are only interested in a single direction (within 5 percent of the mean).
Looking up the Z value for a 99 percent confidence level (one-tailed) in a standard normal distribution table, we can find that Z ≈ 2.326.
Next, we need to calculate the standard deviation (S). The standard deviation is the measure of the spread of the observations around the mean. In this case, we have different standard deviations for each work element:
For work element 1:
Standard deviation (S1) = 0.20 minutes
For work element 2:
Standard deviation (S2) = 0.10 minutes
For work element 3:
Standard deviation (S3) = 0.15 minutes
For work element 4:
Standard deviation (S4) = 0.10 minutes
To determine the overall standard deviation (S), we need to use a formula that takes into account the individual standard deviations and sample sizes of each work element. Since the sample sizes for each work element are not provided, we will assume equal sample sizes for simplicity.
We can calculate the overall standard deviation (S) using the formula for the standard deviation of the sample mean:
S = √(Σ(Si^2) / n)
Where:
Si is the standard deviation for each work element,
Σ represents the summation across all work elements,
and n represents the sample size for each work element (assuming equal sample sizes).
Now, let's calculate the overall standard deviation (S) using the formula and assuming equal sample sizes for each work element:
S = √((S1^2 + S2^2 + S3^2 + S4^2) / n)
Next, we need to convert the desired margin of error (E) into a proportion, which is 5 percent in this case, or 0.05.
Finally, we can substitute all the values into the sample size formula:
n = (Z * S / E)^2
where Z = 2.326, S is the calculated overall standard deviation, and E = 0.05.
By following these steps and substituting the values into the formula, we can determine the required sample size needed to meet the given criteria.