The Dubuque Cement Company packs 80-pound bags of concrete mix. Time-study data for the filling activity are shown in the following table. Because of the high physical demands of the job, the company's policy is a
25%allowance for workers. (Round all intermediate calculations to two decimal places before proceeding with further calculations.
Performance
Observation (seconds)
Element
Rating
1
2
3
4
5
Grasp and place bag
115
%
8
10
8
11
7
Fill bag
85
%
36
40
40
34
110
*
Seal bag
110
%
15
17
14
20
17
Place bag on conveyor
85
%
9
5
8
31
^
36
^
* Bag breaks open; included as delay in the allowance factor
^ Conveyor jams; included as delay in the allowance factor
a) The standard time for this process =
nothing
seconds (round your response to two decimal places).
b) How many observations are necessary for 99% confidence, within 5% accuracy?
The way you have given the data makes it extremely difficult to impossible to answer your question.
To calculate the standard time for this process, we need to calculate the average time per element and multiply it by the total number of elements.
To calculate the average time per element, we need to sum up the time values for each element and divide it by the number of observations.
a) Average time per element:
Grasp and place bag:
(8 + 10 + 8 + 11 + 7) / 5 = 8.8 seconds
Fill bag:
(36 + 40 + 40 + 34 + 110*) / 5 = 52 seconds
Seal bag:
(15 + 17 + 14 + 20 + 17) / 5 = 16.6 seconds
Place bag on conveyor:
(9 + 5 + 8 + 31 + 36^ + 36^) / 6 = 18.3 seconds
*Note: The bag break and conveyor jam elements are not included in the average time calculation.
Total average time per element:
8.8 + 52 + 16.6 + 18.3 = 95.7 seconds
b) To calculate the number of observations necessary for 99% confidence within 5% accuracy, we need to use the following formula:
n = ((Z * σ) / E)^2
Where:
n is the number of observations
Z is the Z score for the desired confidence level (99% = 2.33)
σ is the standard deviation (unknown in this case)
E is the desired accuracy (5% = 0.05)
Since we don't have the standard deviation, we can assume it to be 10% of the average time. Therefore, σ = 0.10 * 95.7 = 9.57 seconds.
Plugging these values into the formula:
n = ((2.33 * 9.57) / 0.05)^2
n = 20,732.10
Therefore, we need a minimum of 20,733 observations (rounded up) for 99% confidence within 5% accuracy.
To find the standard time for this process, we need to calculate the average time for each element of the process and then apply the allowance factor.
a) Calculation of Standard Time:
Step 1: Calculate the average time for each element.
- Grasp and place bag:
Average time = (8 + 10 + 8 + 11 + 7)/5 = 8.8 seconds
- Fill bag:
Average time = (36 + 40 + 40 + 34 + 110)/5 = 52 seconds
(Note: The "*" symbol indicates that there was a delay in one observation due to a bag break, which is not included in the average time calculation.)
- Seal bag:
Average time = (15 + 17 + 14 + 20 + 17)/5 = 16.6 seconds
- Place bag on conveyor:
Average time = (9 + 5 + 8 + 31 + 36)/5 = 17.8 seconds
(Note: The "^" symbol indicates that there were delays in two observations due to conveyor jams, which are not included in the average time calculation.)
Step 2: Apply the performance rating (in percentage) to each average time.
- Grasp and place bag: 115% of 8.8 = 10.12 seconds
- Fill bag: 85% of 52 = 44.2 seconds
- Seal bag: 110% of 16.6 = 18.26 seconds
- Place bag on conveyor: 85% of 17.8 = 15.13 seconds
Step 3: Sum up the adjusted times for each element to get the standard time.
Standard time = 10.12 + 44.2 + 18.26 + 15.13 = 87.71 seconds
Therefore, the standard time for this process is 87.71 seconds.
b) To determine the number of observations necessary for 99% confidence within 5% accuracy, we need to use the formula for calculating the sample size for estimating the population mean:
n = (Z^2 * s^2) / E^2
Where:
n = sample size
Z = z-score corresponding to the desired confidence level (99% confidence corresponds to a z-score of approximately 2.58)
s = standard deviation of the process time (estimate based on previous data or a pilot study)
E = desired margin of error (5% accuracy corresponds to 0.05)
Since the standard deviation of the process time (s) is not provided in the given data, we cannot determine the exact sample size without this information. However, once the standard deviation is known, you can plug in the values into the formula to calculate the required sample size (n).
Note: Time-study data is usually collected from a sample of observations, and the number of observations depends on various factors, including the variability in the data, the desired level of confidence, and the desired level of accuracy.