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 23% allowance for workers. a) Compute the standard time for the bag-packing task. b) How many observations are necessary for 99% confidence, within 5% accuracy? 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 23% allowance for workers. a) Compute the standard time for the bag-packing task. b) How many observations are necessary for 99% confidence, within 5% accuracy? Observations (seconds) Performance

Element 1 2 3 4 5 Rating (%)
Grasp and place bag 8 9 8 11 7 110
Fill bag 36 41 39 35 112a 85
Seal bag 15 17 13 20 18 105
Place bag on conveyor 8 6 9 30b 35b 90

a) To compute the standard time for the bag-packing task, follow these steps:

1. Calculate the average time for each task element:
Grasp and place bag: (8+9+8+11+7)/5 = 8.6 seconds
Fill bag: (36+41+39+35)/4 = 37.75 seconds (we will exclude observation 112 in this calculation as it seems to be an error)
Seal bag: (15+17+13+20+18)/5 = 16.6 seconds
Place bag on conveyor: (8+6+9)/3 = 7.67 seconds (exclude observations 30 and 35 as they seem to be errors)

2. Multiply the average time by the performance rating for each task element:
Grasp and place bag: 8.6 * 1.1 = 9.46 seconds
Fill bag: 37.75 * 0.85 = 32.09 seconds
Seal bag: 16.6 * 1.05 = 17.43 seconds
Place bag on conveyor: 7.67 * 0.9 = 6.9 seconds

3. Add up the times for all task elements:
9.46 + 32.09 + 17.43 + 6.9 = 65.88 seconds

4. Add the 23% allowance for workers:
65.88 * 1.23 = 81.03 seconds

So, the standard time for the bag-packing task is 81.03 seconds.

b) To determine how many observations are necessary for 99% confidence within 5% accuracy, use the following formula:

Number of observations (n) = (Z^2 * SD^2 * p * (1-p)) / E^2

Where:
Z = Z-score for the desired confidence level (2.576 for 99% confidence)
SD = Standard deviation of the time observations
p = Proportion of successes (in this case, successful bag packing)
E = Desired accuracy (0.05)

First, calculate the standard deviation for each task element:
Grasp and place bag: SD = √(((1*(8-8.6)^2)+(1*(9-8.6)^2)+(1*(8-8.6)^2)+(1*(11-8.6)^2)+(1*(7-8.6)^2))/4) = 1.52
Fill bag: SD = √(((1*(36-37.75)^2)+(1*(41-37.75)^2)+(1*(39-37.75)^2)+(1*(35-37.75)^2))/3) = 2.49
Seal bag: SD = √(((1*(15-16.6)^2)+(1*(17-16.6)^2)+(1*(13-16.6)^2)+(1*(20-16.6)^2)+(1*(18-16.6)^2))/4) = 2.41
Place bag on conveyor: SD = √(((1*(8-7.67)^2)+(1*(6-7.67)^2)+(1*(9-7.67)^2))/2) = 1.53

Next, calculate the proportion of successes (p) for each task element based on the performance ratings:
Grasp and place bag: p = 110/110 = 1
Fill bag: p = 85/85 = 1
Seal bag: p = 105/105 = 1
Place bag on conveyor: p = 90/90 = 1

Finally, calculate the number of observations (n) for each task element:

Grasp and place bag: n = ((2.576^2 * 1.52^2 * 1 * (1-1))/(0.05^2)) = 533.84
Fill bag: n = ((2.576^2 * 2.49^2 * 1 * (1-1))/(0.05^2)) = 993.84
Seal bag: n = ((2.576^2 * 2.41^2 * 1 * (1-1))/(0.05^2)) = 910.66
Place bag on conveyor: n = ((2.576^2 * 1.53^2 * 1 * (1-1))/(0.05^2)) = 583.83

Since the Fill bag task has the highest number of observations necessary for 99% confidence within 5% accuracy, we will use that value as the answer: 994 observations are required (rounded up to the nearest whole number).

To compute the standard time for the bag-packing task, we need to follow these steps:

Step 1: Calculate the average observed time for each element.
Element 1 (Grasp and place bag): (8 + 9 + 8 + 11 + 7) / 5 = 8.6 seconds
Element 2 (Fill bag): (36 + 41 + 39 + 35) / 4 = 37.75 seconds
Element 3 (Seal bag): (15 + 17 + 13 + 20 + 18) / 5 = 16.6 seconds
Element 4 (Place bag on conveyor): (8 + 6 + 9 + 30 + 35) / 5 = 17.6 seconds

Step 2: Calculate the average observed time for the bag-packing task.
Average observed time = Element 1 + Element 2 + Element 3 + Element 4 = 8.6 + 37.75 + 16.6 + 17.6 = 80.55 seconds

Step 3: Apply the allowance factor.
Standard time = Average observed time / (1 - Allowance)
Standard time = 80.55 / (1 - 0.23) = 104.87 seconds

Therefore, the standard time for the bag-packing task is 104.87 seconds.

To calculate the number of observations necessary for 99% confidence within 5% accuracy, we can use the following formula:
n = (Z * Z * p * (1 - p)) / (E * E)
where:
n = sample size
Z = Z-score for the desired confidence level (99% confidence corresponds to a Z-score of 2.58)
p = estimated proportion of the population (0.5 for maximum variability)
E = desired level of accuracy (5% accuracy corresponds to 0.05)

Plugging in the values, we have:
n = (2.58 * 2.58 * 0.5 * (1 - 0.5)) / (0.05 * 0.05)
n = 334.28

Since the sample size should be a whole number, we round up to the nearest whole number. Therefore, the number of observations necessary for 99% confidence within 5% accuracy is 335.

To compute the standard time for the bag-packing task, we need to follow the formula:

Standard time = Total observed time / (1 + Allowance factor)

a) From the given data, we need to calculate the total observed time for the bag-packing task.

Total observed time = Sum of observation times for each performance element

Total observed time = (Fill bag) + (Seal bag) + (Place bag on conveyor) = 112a + (15 + 17 + 13 + 20 + 18) + (8 + 6 + 9 + 30b + 35b)

Note: The values marked with 'a' and 'b' need to be excluded as outliers since they significantly differ from the other observations.

Total observed time = 112 + (15 + 17 + 13 + 20 + 18) + (8 + 6 + 9) [excluding 'a' and 'b']

Total observed time = 112 + 83 + 23 = 218 seconds

Next, we need to calculate the allowance factor.

Allowance factor = Allowance / 100

Given that the company's policy is a 23% allowance for workers, the allowance factor is:

Allowance factor = 23 / 100 = 0.23

Now we can compute the standard time for the bag-packing task:

Standard time = Total observed time / (1 + Allowance factor)

Standard time = 218 / (1 + 0.23) = 218 / 1.23 ≈ 177.24 seconds

Therefore, the standard time for the bag-packing task is approximately 177.24 seconds.

b) To determine the number of observations necessary for 99% confidence within 5% accuracy, we need to use the following formula:

Number of observations = (Z-score)^2 * (Standard deviation)^2 / (desired error)^2

The Z-score is obtained from the standard normal distribution table corresponding to the desired confidence level. For 99% confidence, the Z-score would be approximately 2.33.

The desired error is given as 5% accuracy, which can be expressed as 0.05.

To calculate the standard deviation, we need to find the sample standard deviation from the given data:

Sample standard deviation = sqrt((sum of (observation time - mean)^2) / (n - 1))

First, find the mean of the observation times:

Mean = (8 + 9 + 8 + 11 + 7 + 36 + 41 + 39 + 35 + 15 + 17 + 13 + 20 + 18 + 8 + 6 + 9) / 17 = 276 / 17 ≈ 16.24

Next, calculate the sum of the squared differences between each observation time and the mean:

Sum of (observation time - mean)^2 = (8 - 16.24)^2 + (9 - 16.24)^2 + (8 - 16.24)^2 + (11 - 16.24)^2 + (7 - 16.24)^2 + (36 - 16.24)^2 + (41 - 16.24)^2 + (39 - 16.24)^2 + (35 - 16.24)^2 + (15 - 16.24)^2 + (17 - 16.24)^2 + (13 - 16.24)^2 + (20 - 16.24)^2 + (18 - 16.24)^2 + (8 - 16.24)^2 + (6 - 16.24)^2 + (9 - 16.24)^2

Sum of (observation time - mean)^2 = 221.64

Finally, calculate the sample standard deviation:

Sample standard deviation = sqrt(221.64 / (17 - 1)) ≈ 4.09

Now we can compute the number of observations:

Number of observations = (Z-score)^2 * (Standard deviation)^2 / (desired error)^2

Number of observations = (2.33)^2 * (4.09)^2 / (0.05)^2 ≈ 853.96

Therefore, approximately 854 observations are necessary for 99% confidence, within 5% accuracy.

Note: These calculations assume that the data provided represents a random and representative sample.