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In the context of statistics, a point estimate is a single value used to approximate a population parameter, like the mean or average of a dataset. In our scenario, the point estimate is particularly crucial, as it provides an immediate, direct guess of the overall figure we aim to understand or predict.

For the exercise in question, the point estimate for the population mean is represented by the sample mean, denoted as \( \bar{x} \). This sample mean is derived from the data collected during a study. In this example, the sample mean \( \bar{x} \) is given as 20. Thus, the point estimate for the population mean number of dollars that steady smokers spend per week on cigarettes is 20.

What does this signify? It indicates that based on the surveyed sample, it is estimated that, on average, steady smokers spend \( \$20 \) a week on cigarettes. This figure serves as a central component when planning further analyses or making predictions about the entire population of steady smokers.

The standard error (SE) is a critical metric in determining how accurately the sample mean represents the population mean. It quantifies the dispersion or variability of the sample mean estimates around the true population mean. Essentially, it gives us a sense of how much the sample mean might fluctuate, if different samples were taken from the population.

To calculate the standard error, we use the formula \( SE = \frac{\sigma}{\sqrt{n}} \) where \( \sigma \) is the standard deviation, and \( n \) is the sample size. In our case, with a standard deviation of \(5\) and a sample size of \(49\), the calculation becomes \( SE = \frac{5}{\sqrt{49}} = \frac{5}{7} \approx 0.7143 \).

A smaller standard error means the sample mean is a more accurate reflection of the actual population mean. This is why larger sample sizes and smaller standard deviations generally lead to a more precise estimation of the population mean.

The margin of error (ME) is an integral part of creating a confidence interval; it showcases the degree of uncertainty we have regarding our point estimate. It essentially outlines the range within which the true population mean likely falls, given our confidence level.

The formula to determine the margin of error is \( ME = z \times SE \), where \( z \) is the z-score or critical value linked to the confidence level and \( SE \) represents the standard error. In the context of a 95% confidence interval, the critical z-value is about 1.96. Utilizing our already calculated SE of approximately 0.7143, we compute \( ME = 1.96 \times 0.7143 \approx 1.4 \).

This margin of error, when added to and subtracted from the point estimate, provides the boundaries of our confidence interval. As such, it helps us confidently state that the true mean value lies within a specified range around our point estimate, reflecting our level of certainty in these estimates.