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In statistics, the term "sample size" refers to the number of observations or data points collected in a study, which is represented by the variable \( n \). It is a crucial element in determining the reliability and accuracy of your statistical inferences.
An appropriate sample size helps ensure that the sample data properly represents the larger population.
When the sample size is large, the estimated statistics (like the sample mean and variance) tend to approximate the true population parameters more closely. In our exercise, a sample size of 250 was used to estimate the average job tenure of all workers aged 16 and older. Because this is a large sample, we can use the normal distribution approximation to calculate the confidence interval effectively.
Choosing a larger sample size can reduce sampling error, enhancing the precision of your estimate. However, larger samples may be more costly or time-consuming to collect, so there's always a balance to strike between precision and resources.

The standard error quantifies the amount of variability in the sample mean from the true population mean. It is a critical component in the construction of confidence intervals.
The standard error is calculated using the formula \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
In our scenario, with a population standard deviation of 3.8 years and a sample size of 250, the standard error is approximately 0.24. This indicates that the average job tenure from different samples would deviate by about 0.24 years from the true population mean.
A smaller standard error means that our sample mean is more tightly clustered around the population mean, leading to narrower confidence intervals and more precise estimates. Lowering the standard deviation or increasing the sample size can help achieve a smaller standard error.

The z-score is a standard score that indicates how many standard deviations an element is from the mean. When constructing confidence intervals, the z-score (represented as \( z^* \)) is used to find critical values based on the preferred level of confidence.
For example, a 99.9% confidence level corresponds to a critical z-score of approximately 3.291, as shown in the exercise. This value means that there is only a 0.1% probability that the true population mean falls outside the interval when constructing a 99.9% confidence interval.
You determine the critical z-score using a standard normal distribution table, which lists the cumulative probabilities of a standard normal distribution.
Utilizing the z-score, you can calculate the margin of error and ultimately the confidence interval. Higher confidence levels demand higher z-scores, resulting in wider intervals to ensure the true parameter is captured within the interval.