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The sample mean, often denoted as \( \bar{x} \), plays a crucial role in statistical analysis. Imagine you want to know the average amount people typically spend on graduation gifts. Instead of asking everybody, because that would be impossible, you ask a smaller group - a sample. The sample mean is the average of this smaller group, and it represents the best guess of the overall population's mean.In the provided exercise, the sample mean was \( \\(55.05 \). This indicates that, based on the surveyed group, the average expenditure on graduation gifts in 2007 was approximately \( \\)55.05 \) per gift. It's a starting point for further analysis, especially when we need to make inferences about the entire population.

Standard deviation is a measure of how spread out the numbers in a data set are. It's like a ruler showing how much variation exists from the average, or mean. A low standard deviation means that most of the numbers are close to the average, while a high standard deviation indicates a wider spread.In the context of the exercise, the standard deviation was \( \\(20 \). This suggests that the amount spent on graduation gifts varied by about \( \\)20 \) from the average of \( \$55.05 \). When creating a confidence interval, a smaller standard deviation generally leads to a more precise interval, as the data points are closer to the mean.

The Z-distribution, also known as the standard normal distribution, is a cornerstone of statistical analysis. It is a symmetrical bell-shaped curve that shows the distribution of data points in a normal dataset. This distribution becomes crucial when dealing with large sample sizes. For our exercise, since the sample size of 2815 is significantly large, we justify our use of the Z-distribution to calculate the confidence interval. The Z-distribution allows us to estimate the range in which the true population mean lies, when the population standard deviation is unknown but the sample size is suitably large. For a 98% confidence interval, the Z-value used is approximately 2.33, found using a standard Z-score table or calculator. This value tells us about the probability of the mean falling within a particular range.

The margin of error is what gives us the wiggle room around our sample mean, making it possible to create a confident range for the true mean. It is essentially the little buffer that accounts for any sampling variability. The margin of error can be calculated using the formula: \[ E = Z \times \left( \frac{s}{\sqrt{n}} \right) \] where \( Z \) is the Z-value, \( s \) is the standard deviation, and \( n \) is the sample size.In our exercise on graduation gifts, the margin of error allows us to say that while the average spending was \( \$55.05 \), thanks to the margin of error, the true average spending could be a bit lower or higher. After performing the calculation using \( Z = 2.33 \), \( s = 20 \), and \( n = 2815 \), you gain this crucial information that helps define the confidence interval.