91Ó°ÊÓ

The population mean is a central element of statistics that represents the average of a complete set of observations in a population. When conducting surveys or experiments, we often gather data from a smaller sample instead of the entire population. This sample provides an estimate of the population mean. In our case, the survey found a sample mean of 26 weeks, suggesting that the average time it takes for laid-off executives to find a new position is around 26 weeks. However, since we've only surveyed 50 executives, this is an estimate and not the exact population mean.

To construct a more reliable estimate, we employ the concept of a confidence interval. It helps us understand in what range the actual population mean is likely to lie, given our sample mean and characteristics.

The standard error (SE) is a measure that provides insight into the precision of the sample mean as an estimate of the population mean. It tells us how much variability we can expect in our sample means if we were to take multiple samples from the same population. The SE is computed using the formula: \[ SE = \frac{s}{\sqrt{n}} \] where \(s\) is the sample standard deviation (6.2 weeks in this case), and \(n\) is the sample size (50 executives in our example). In our survey, the resulting standard error was approximately 0.876 weeks.

This small number indicates that the sample mean of 26 weeks is a fairly precise estimate of the population mean.

The margin of error (MOE) is crucial in determining how much we can trust our sample mean as an estimate for the population mean. It provides a range that shows how much the true population mean might differ from our sample mean. We calculate it by multiplying the Z-score by the standard error: \[ MOE = Z \times SE \] In most cases, the Z-score used for a 95% confidence interval is 1.96. Using our standard error (0.876), the margin of error comes out to approximately 1.718 weeks.

This MOE tells us that we can expect the true population mean to likely be within 1.718 weeks above or below our sample mean of 26 weeks.

The Z-score plays an essential role in constructing confidence intervals by providing a multiplier for the standard error. It translates a desired coverage probability into a standardized number that indicates how many standard errors away from the mean corresponds to the confidence interval bounds. For a 95% confidence interval, this Z-score is typically 1.96.

By using this Z-score, we can calculate the range within which we expect the population mean to fall. This standardized score ensures that the interval covers approximately 95% of possible population means in repeated sampling, highlighting its relevance in inferential statistics.