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When we have data from a sample rather than an entire population, especially when the sample size is small (less than 30), we use the T-Distribution. This is necessary because, unlike the normal distribution, the T-Distribution is more spread out or has "fatter tails." These tails account for the added variability expected with smaller sample sizes. The T-Distribution adjusts for the fact that we estimate the population standard deviation from a small sample.
This means you are less sure about where the true population mean might fall, so the confidence interval is wider. The T-Distribution becomes increasingly similar to the normal distribution as the sample size grows. It's good to think of it as a more conservative estimate that compensates for limited data.

The sample mean is an estimate of the population mean. In a sample, the mean is denoted as \[ \bar{x} \]. It's calculated by adding up all the data values and dividing by the number of observations in the sample.
In the problem, the sample mean, \[ \bar{x} \], is given as 14,381. This value is used as the midpoint of our confidence interval, serving as the best estimate of the true population mean based on the data from the sample.
Always remember that the sample mean is just an estimate and can differ from the true population mean when different samples are taken.

Standard Error (SE) considers how much discrepancy exists between the sample mean and the actual population mean. It represents the variability of sample means we might expect if we took multiple samples from the same population.
The formula for standard error is:\[ SE = \frac{s}{\sqrt{n}} \], where \[ s \] is the sample standard deviation and \[ n \] is the sample size. The standard error decreases as the sample size increases because there is less variability expected with larger samples.
In our study, the SE was calculated to be approximately 378.4. This value helps quantify the precision of the sample mean as an estimate of the population mean.

Degrees of freedom (df) is a concept that refers to the number of independent values that can vary in a calculation. In the context of calculating the T-Distribution, especially for a confidence interval, degrees of freedom is given by the formula:\[ df = n - 1 \], where \[ n \] is the sample size.
For a sample size of 25, we would have 24 degrees of freedom. This value is crucial when using a t-table to find the critical t-value needed to construct a confidence interval.Degrees of freedom are important because they affect the shape of the T-Distribution, especially with smaller sample sizes. The fewer the degrees of freedom, the thicker the tails of the T-Distribution, indicating greater uncertainty in the estimate of the population mean.