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The sample mean, often denoted as \( \bar{x} \), represents the average value from a set of samples. In statistical terms, it is the sum of all observations divided by the number of observations in the sample. For example, if you have collected data from 200 savings account increases and calculated that their mean is 7.2%, this 7.2% is your sample mean.
The sample mean provides a useful estimate of the population mean, which is the average you would expect if you could measure the entire population. However, because it is based on a sample, this mean is only an estimate and has some associated uncertainty.
Understanding the sample mean is crucial because it forms the basis of constructing a confidence interval, which describes the range within which we might expect the true population mean to fall.

The standard error (SE) measures the variability or dispersion of the sample mean. It describes how much the sample mean would vary from one sample to another if you were to repeat your sampling procedure many times. The formula for the standard error of the mean is:

• \( SE = \frac{s}{\sqrt{n}} \)

where \( s \) is the sample standard deviation and \( n \) is the sample size.
In our case with a sample size of 200 and a standard deviation of 5.6%, the standard error is about 0.396%.
The standard error is crucial for constructing a confidence interval, as it helps you determine the margin of error, showing how much your sample mean can reasonably vary within the population mean.

The margin of error (ME) quantifies the range of values below and above your sample mean that will likely include the true population mean. It reflects how precise your estimate is, taking into account variability due to the sample size and sample standard deviation.
The margin of error is computed as follows:

• \( ME = z \times SE \)

where \( z \) is the critical value from the standard normal distribution for the desired confidence level, and \( SE \) is the standard error.
For instance, with a 95% confidence level, the critical value \( z \) is typically 1.96. Using the previous calculation, this results in a margin of error of approximately 0.776%.
This means that the true mean increase in the community's savings accounts is expected to lie within 0.776% above or below the calculated sample mean (7.2%).

The critical value is a factor used to calculate the margin of error and, consequently, the confidence interval. It is determined by the desired confidence level, which reflects how certain you want to be that the interval contains the true population mean.
For most practical purposes, when the sample size is large, the critical value is derived from the standard normal distribution. For instance, a 95% confidence level, which is common in statistics, corresponds to a critical value (\( z \)) of approximately 1.96. This critical value indicates the number of standard deviations a data point must be from the mean for the data to be considered sufficiently close within the desired confidence level.
Understanding the critical value is crucial, as it helps balance your need for precision (small margin of error) against your confidence level (how sure you are of your estimate). The critical value, combined with the standard error, shapes the ultimate size of the confidence interval.