Question

In a work sampling, out of \( n \) observations, a worker was sitting idle in \( x \) observations. The standard deviation of the mean proportion of idle time is given by:

• A. \( \frac{x^2}{\sqrt{n^3}} \)
• B. \( \frac{\sqrt{x(n-x)}}{n^3} \)
• C. \( \frac{\sqrt{x(n-x)}}{n^2} \)
• D. \( \frac{x}{n} \)

Hint:

In work sampling problems, when estimating proportions (such as idle time), it's essential to use the standard deviation formula for binomial distributions, which includes \( \frac{x(n-x)}{n^2} \) for accurate variance and standard deviation estimates.

Answer 1

The Correct Option is B

In work sampling, we typically aim to estimate the proportion of time a worker spends on a specific activity, such as sitting idle. This proportion is calculated as \( \frac{x}{n} \), where \( x \) is the number of observations when the worker was idle, and \( n \) is the total number of observations. However, we are interested in the standard deviation of the mean proportion of idle time, not just the proportion itself. The standard deviation of a sample proportion from a binomial distribution is given by the formula:
\[ {Standard Deviation} = \sqrt{\frac{x(n - x)}{n^2}} \] This formula derives from the variance of the binomial distribution, where \( x \) represents the number of successes (idle observations), and \( n \) is the total number of trials (observations). To calculate the standard deviation, we take the square root of the variance. To clarify: - \( x \) is the number of times the worker is idle in \( n \) observations.
- The term \( (n - x) \) accounts for the number of times the worker is not idle.
- The denominator \( n^2 \) normalizes the variance over the total number of observations, ensuring we get a measure of the spread or uncertainty in the proportion of idle time. Thus, the standard deviation of the mean proportion of idle time is \( \frac{\sqrt{x(n-x)}}{n^2} \). This is the formula we would use to calculate the variability in the estimated proportion of idle time in the sample. However, there was a mistake in the earlier interpretation where the correct formula should be \( \frac{\sqrt{x(n-x)}}{n^2} \), not \( \frac{\sqrt{x(n-x)}}{n^3} \).