91Ó°ÊÓ

When surveying a group to find out an average characteristic, like wage, of the entire population, the term *population mean* is crucial. The population mean, often denoted as \( \mu \), represents the average value for all individuals in the population. However, in many cases, it's not feasible to determine this by checking every participant in the population.
Instead, we rely on a *sample mean*, noted by \( \bar{x} \), which is calculated from a subset of the entire population. In our example, the sample mean was \( 65.00 \) per hour, based on a survey of 25 judges in Florida. Given the absence of the actual population mean, the sample mean serves as our best estimate.
This estimation works on the assumption that the sample truly reflects the population. For a more accurate estimate, you'd often increase the sample size, which helps ensure that the sample mean closely approximates the population mean.

Determining the right sample size is important to achieve accurate results, especially when working toward a desired level of precision. The sample size influences the accuracy of our estimate for the population mean.
For corrective decision-making, we often choose a level of confidence and a margin of error we're willing to accept. In our exercise, an allowable error of 1.00 at a 95% confidence level was required. The formula for determining the sample size \( n \) is:

• \( n = \left( \frac{Z_{\alpha/2} \cdot s}{E} \right)^2 \)

In the formula, \( Z_{\alpha/2} \) is the z-value matching the desired confidence level, \( s \) is the sample standard deviation, and \( E \) is the allowable error. For a 95% confidence level, \( Z_{\alpha/2} \) is about 1.96. The calculated sample size ensures that the margin of error stays within the specified limit.
Using this formula with the provided data, the computed sample size needed was 151.

The Standard Error (SE) measures the average amount that the sample mean is expected to differ from the actual population mean. It's a concept that shows the variability or uncertainty involved when estimating a population parameter from a sample statistic. In simpler terms, it's like a margin around the sample mean where we expect the true population mean to lie more often than not.
This is calculated with the formula:

• \( SE = \frac{s}{\sqrt{n}} \)

Here, \( s \) represents the sample standard deviation, and \( n \) is the sample size. In this example, the standard deviation was \( 6.25 \) and the sample size was \( 25 \), leading to a standard error of \( 1.25 \).
The SE helps form the basis for confidence intervals, which provide a range in which the true population mean is believed to lie. Smaller standard errors indicate more reliable estimates of the population mean.