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The **sample mean** provides a measure of the central tendency of a set of data. It's a critical component in statistics, especially when dealing with continuous populations. The sample mean is represented by the symbol \( \bar{x} \). In essence, this value is the average of all measurements in your sample. It's calculated using the formula: \[\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\]Where \( n \) is the sample size, and \( x_i \) represents each individual observation in the sample. The sample mean helps in making inferences about the population mean. In the exercise, the sample mean was given as 32.0, derived from a sample of 64 observations. This value acts as a point estimate of the population mean when building the confidence interval.

The **population standard deviation** is a measure reflecting the amount of variation or dispersion in a population. It indicates how much individual data points deviate from the population mean. It's denoted by the symbol \( \sigma \). A smaller standard deviation means the data points tend to be close to the mean, while a larger one indicates a wider spread. To find the standard deviation of a sample mean (often called the standard error), we divide the population standard deviation by the square root of the sample size:\[\sigma_{x} = \frac{\sigma}{\sqrt{n}}\]In the exercise, the population standard deviation was provided as 2.4, which allowed us to calculate the standard error of the sample mean as 0.3 when applying the formula. Understanding this helps in gauging the accuracy of the sample mean as an estimate of the population mean.

The **Z-score** is a statistical measurement that describes a data point's relation to the mean of the group of data. It's used to signify how many standard deviations a data point is from the mean. When constructing confidence intervals, the Z-score corresponding to the desired confidence level is critical.For example, with a confidence level of 90%, the Z-score can be found using statistical tables, which is denoted by \( Z(\alpha/2) \). For a 90% confidence interval, this Z-score is 1.645. This value derives from a standard normal distribution and indicates that there is a 5% probability on either side of the distribution tails. Having the correct Z-score is vital for calculating the confidence interval accurately.

The **error of estimate** (E) signifies the maximum expected difference between the sample mean and the true population mean. It's an essential part of confidence interval calculations as it determines the precision of the estimate. The error of estimate can be found using the formula:\[E = Z(\alpha/2) \times \sigma_{x}\]Here, \( Z(\alpha/2) \) is the Z-score associated with the confidence level, and \( \sigma_{x} \) is the standard deviation of the sample mean. In our example, the error of estimate calculated was approximately 0.492. This means the true population mean is likely to be within 0.492 units from the sample mean on either side, thereby shaping the width of the confidence interval.