A job shop manager wants to determine the proportion of time the workers are idle. The shop employs two types of workers. Type 1 includes three workers and Type 2 includes two workers. The manager suspects that the proportion of idle time is about 20 percent for Type 1 workers and 15 percent for Type 2 workers. If the manager wants a 95 percent confidence level (z = 1.96) and a degree of precision of + 0.05, how many trips will the observers have to take through the group?
A) fewer than or equal to 80
B) greater than 80 but fewer than or equal to 90
C) greater than 90 but fewer than or equal to 100
D) greater than 100

Question

Quizplus · Accepted Answer

We need to calculate the sample size based on the precision, confidence level, and the estimated proportions of idle time for each worker type.

For Type 1 workers:
- Estimated proportion of idle time = 0.20
- Estimated proportion of work time = 0.80
- Number of workers = 3

Using the formula:
n = (Z^2 * p * q * N) / [(Z^2 * p * q * N) + (N - 1) * d^2]

where:
- Z = 1.96 (for 95% confidence level)
- p = 0.20
- q = 0.80
- N = 3 (number of workers)
- d = 0.05 (degree of precision)

Plugging in the values, we get:
n = (1.96^2 * 0.20 * 0.80 * 3) / [(1.96^2 * 0.20 * 0.80 * 3) + (3 - 1) * 0.05^2] = 21.16

Round up to the nearest whole number, we get n = 22

For Type 2 workers:
- Estimated proportion of idle time = 0.15
- Estimated proportion of work time = 0.85
- Number of workers = 2

Using the same formula, we get:
n = (1.96^2 * 0.15 * 0.85 * 2) / [(1.96^2 * 0.15 * 0.85 * 2) + (2 - 1) * 0.05^2] = 19.89

Round up to the nearest whole number, we get n = 20

Therefore, the total number of trips needed is:
Total trips = n1 + n2 = 22 + 20 = 42

This is less than or equal to 100, but greater than 90, so the answer is C.

A Job Shop Manager Wants to Determine the Proportion of Time