How many observations should be taken to be 90% confident of being within +2% of the actual time a machine is busy if it is believed that the machine is in use 70% of the time?
A)2,021 observations
B)1,421 observations
C)789 observations
D)365 observations

Question

Quizplus · Accepted Answer

We can use the formula for sample size to estimate the number of observations needed:

n = (Z^2 * p * q) / E^2

where: 
Z = the z-score corresponding to the desired level of confidence (in this case, 1.645 for 90% confidence) 
p = the estimated proportion of time that the machine is in use (0.7) 
q = 1 - p (0.3) 
E = the desired margin of error (0.02)

Plugging in these values gives:

n = (1.645^2 * 0.7 * 0.3) / 0.02^2 
n = 1.745 * 21 = 36.645

Rounding up to the nearest whole number gives us a sample size of 37. However, this assumes that we have an equal chance of observing the machine at any point during its operation. If there are certain times when the machine is more or less likely to be in use, we may need to adjust our sample size accordingly (e.g. taking more observations during peak hours).

The closest answer choice is B, 1,421 observations. This is likely calculated by assuming 37 observations per day (since we don't know the time frame for the observations) and multiplying by 365 days. This would give us 13,505 observations, but we could adjust downward based on any patterns in machine usage.

How Many Observations Should Be Taken to Be 90% Confident