91Ó°ÊÓ

The standard error (SE) is a statistical term that measures the accuracy with which a sample represents a population. In other words, it quantifies the variability of the sampling distribution of a statistic, most commonly of the mean. The formula to calculate the standard error is:
\[\begin{equation}SE = \frac{\sigma}{\sqrt{n}}\end{equation}\]
where \(\sigma\) is the standard deviation of the population and \(n\) is the sample size. When the population standard deviation is unknown, as is often the case, the sample standard deviation (denoted as \(s\)) is used instead, yielding the formula:
\[\begin{equation}SE = \frac{s}{\sqrt{n}}\end{equation}\]
In the given exercise, the standard error is calculated using this formula with the provided sample standard deviation and size, which helps to determine how much the sample mean \(\bar{x}\) is expected to fluctuate from the true population mean \(\mu\).

The Z-score, also known as a standard score, quantifies how many standard deviations an element is from the mean of a distribution. It is a pivotal concept when working with normal distributions. The Z-score can be calculated by the formula:
\[\begin{equation}Z = \frac{(X - \mu)}{\sigma}\end{equation}\]
where \(X\) is the value from the distribution, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation. In the context of confidence intervals, the Z-score provides a critical value that determines the range around the sample mean within which the true population mean is likely to lie with a given level of confidence. Higher confidence levels correspond to larger Z-scores. For our example, a 99% confidence level leads to a Z-score of approximately 2.576, indicating that the interval extends 2.576 standard errors from the sample mean on both sides to capture the true population mean.

The normal distribution, commonly associated with the bell curve, is a continuous probability distribution that is symmetrical around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In real-life situations, the normal distribution can be used to describe characteristics such as test scores, measurement errors, and many other phenomena.When constructing confidence intervals, the normal distribution assumes that the means of the samples follow a normal distribution around the true population mean. This allows us to make probabilistic statements about where the true mean lies based on our sample data. A 99% confidence interval indicates that we are 99% certain that the true population mean falls within our calculated range. The