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In statistics, when we deal with smaller sample sizes, typically less than 30, we use the t-distribution instead of the normal distribution. The t-distribution is a type of probability distribution that is symmetric, similar to the normal distribution, but has heavier tails. This means it is more prone to producing values that fall far from its mean.

Why use the t-distribution? Because it accounts for the extra variability that can occur with smaller sample sizes, providing a more accurate critical value for calculating confidence intervals. The critical value can be found using a t-table or statistical software based on the desired confidence level and degrees of freedom, defined as the sample size minus one (i.e., \( df = n - 1 \)).

In our exercise, the sample size was 25, leading to 24 degrees of freedom. For a 90% confidence level, the critical value was approximately 1.711. This critical value is key when calculating the margin of error for the confidence interval.

The standard error (SE) is a crucial component in estimating how much the sample mean is expected to vary from the true population mean. It tells us the precision of the sample mean estimate.

Standard error is calculated using the formula:

• \( SE = \frac{s}{\sqrt{n}} \)

where \( s \) is the sample standard deviation, and \( n \) is the sample size.

In our example, with a standard deviation of 7,568 and a sample size of 25, the standard error calculates to \( SE = 1,513.6 \). This figure serves as an indicator of how much disparity there can be from one sample to another, or from the sample mean to the true population mean.

The margin of error (E) expresses the range within which we expect the population mean to fall. It's a critical aspect of constructing a confidence interval, which ultimately helps us make inferences about the population based on sample data.

The margin of error is calculated using the formula:

• \(E = t_{\alpha/2} \times SE \)

where \( t_{\alpha/2} \) is the critical value from the t-distribution, and SE is the standard error.

In the exercise, with \( t_{\alpha/2} \approx 1.711 \) and \( SE = 1,513.6 \), the margin of error is \( E \approx 2,589.1 \). We then use this margin to create the confidence interval, determining that the true population mean lies within 2,589.1 units of the sample mean.

The sample mean, denoted as \( \bar{x} \), is the average of all measurements in a sample. It is a simple yet powerful statistic used to estimate the population mean. The sample mean serves as the center of the confidence interval.

In our given exercise, the sample mean was 55,051. This value provides a starting point for our calculations and helps in constructing the confidence interval.

Though the sample mean is a robust estimate of the population mean, it does not account for variability among samples. That's why we consider the standard error and margin of error to provide a range (or interval) where the population mean likely resides.