Question

A company wishes to determine the proportion of time workers are idle. Using work sampling, the idle time during 150 observations is 26%. If the company wants a 95% confidence interval, how many more observations are needed if the company wants to be within +/-4% of the true proportion?
1. 9 observations
2. 150 observations
3. 312 observations
4. 462 observations

Answer 1

Correct Answer - Option 3 : 312 observations

Concept:

Work Sampling:

Work sampling is a method in which a large number of instantaneous observations are made at random time intervals over a period of time or a group of machines, workers or processes/operations. Each observation records what is happening at that instant and the percentage of observations recorded for a particular activity or delay/idleness is a measure of the percentage of time during which that activity or delay/idleness occurs.

Sample Size:

\(n = \frac{{{Z^2}p\left( {1 - p} \right)}}{{{e^2}}}\)

Where, n = Sample size, Z = Z value (e.g. 1.96 for 95% confidence level), p = percentage expressed as decimal, e = Acceptable error percentage as a decimal (e.g., 0.04 = ± 4%)

Confidence	Z
99.9%	3.250
99%	2.326
95%	1.960
90%	1.645

 

Calculation:

Given:

p = 0.26, Old observation (n₁) = 150, e = 0.04 = ± 4%, Z = 1.960 (for 95% confidence level) 

Observation needed (n₂) for 95% confidence level is

\({n_2} = \frac{{{Z^2}p\left( {1 - p} \right)}}{{{e^2}}}\)

\({n_2} = \frac{{{{\left( {1.960} \right)}^2} \times 0.26 \times \left( {1 - 0.26} \right)}}{{{{\left( {0.04} \right)}^2}}}\)

n₂ = 461.9 ≈ 462 Observation

Change in observation is = (462 - 150) = 312

So, 312 more observations are needed to meet the requirement.