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The sample mean is a crucial concept when it comes to estimating population parameters because it provides the best point estimate for a population mean, denoted as \( \mu \). In the context of data analysis, the sample mean, \( \bar{x} \), is simply the average of your sampled data points. It can be calculated by summing all the sample values and then dividing by the number of values in the sample.

For example, in the case of finding a 95% confidence interval as given, the sample mean is \( 84.6 \). This means that if you take the average of all the data points in our sample, the result is exactly \( 84.6 \).
This is your central figure, from which the confidence interval, or range where the true population mean is expected to lie, is developed. By itself, the sample mean gives you a glimpse of the population's central tendency. However, to understand how precise this estimate is, we need to delve further into calculating margins of error and standard error.

The margin of error is a key component in determining the confidence interval around a sample mean. It reflects the extent of possible variation or uncertainty in the estimation of the population mean. Essentially, the margin of error offers a buffer around your sample mean, suggesting that the true population mean is likely contained within this range.

For calculating the margin of error, you need to consider both the t-score and the standard error. In the provided example, a t-score of approximately 2.021 is used. The t-score corresponds to the confidence level you're interested in (such as 95%) and the degrees of freedom associated with your sample size.
The margin of error (\( ME \)) is then calculated using the formula:

• \( ME = t \times SE \)

where \( SE \) represents the standard error. In this exercise, putting all components together, we compute \( ME = 2.021 \times 1.202 \), which yields approximately \( 2.43 \). This means we add and subtract this margin from the sample mean, framing the confidence interval within which the true population mean likely falls.

The standard error is a measure that gives insight into how much the sample mean is expected to fluctuate from the true population mean. It is essentially the standard deviation of the sampling distribution of the sample mean. A smaller standard error suggests that the sample mean is a more precise estimator of the population mean.

To compute the standard error, you use the formula:

• \( SE = \frac{s}{\sqrt{n}} \)

where \( s \) is the sample standard deviation and \( n \) is the sample size. In our exercise, the sample standard deviation is given as \( 7.8 \) and the sample size \( n \) is \( 42 \). Using these values in our formula gives:

• \( SE = \frac{7.8}{\sqrt{42}} \approx 1.202 \)

This calculation shows that the sample mean can be off from the true mean by about \( 1.202 \). Understanding the standard error is imperative as it directly impacts the width of the confidence interval: less standard error narrows your confidence interval, leading to greater precision.