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When working with small sample sizes or when the population standard deviation is unknown, the t-distribution is used instead of the normal distribution. This distribution accounts for the smaller sample size by having thicker tails, which means there's a higher probability of values far from the mean.
For this exercise, the sample size is 50, which is fairly large but often the t-distribution is still used in practice for sample sizes under 30. However, since the exact population variance remains unknown, it is still appropriate for larger samples.
The t-distribution is applicable when dealing with sample means and becomes more similar to the normal distribution as the sample size increases. It is defined by its degrees of freedom, which are directly related to the sample size.

The margin of error (ME) quantifies the amount of random sampling error in a survey's results. It tells us how precise our estimate of the population parameter is.
For a confidence interval, the margin of error is added to and subtracted from the sample mean to find the upper and lower bounds of the interval.
In this problem, the formula to calculate the margin of error is:

• ME = \( t^* \times \frac{s_{\text{diff}}}{\sqrt{n}} \)

Here, \( t^* \) is the critical value from the t-distribution for the desired confidence level, \( s_{\text{diff}} \) is the standard deviation of the sample differences, and \( n \) is the sample size. This calculation provides the ME value, which is then used to construct the confidence interval around the sample mean.

Degrees of freedom (df) in statistics are the number of values in a calculation that are free to vary. They are crucial for deriving the critical value of a statistic, such as \( t^* \) in the t-distribution.
For a sample, df is typically the sample size minus one:

• df = n - 1

In our tree example, calculating degrees of freedom involves the sample size, which is 50. Thus, the df = 49.
The degrees of freedom impact the shape of the t-distribution, especially how spread out it is. Fewer degrees of freedom result in a curve that is more spread out, which reflects greater uncertainty in estimating the population parameter.

A sample mean (\( \bar{x} \)) is the average of a set of observations from a sample, representing the center of the dataset. It serves as an estimate for the population mean when the population mean is unknown.
In this exercise, the sample mean difference, denoted as \( \bar{x}_{\text{diff}} \), is 12.5 feet. This value represents the average growth between 2009 and 2019 for the selected trees.
The sample mean is fundamental when calculating confidence intervals, such as the one estimated here, as it forms the basis around which the interval is constructed. Its accuracy is dependent on having a representative sample.

Standard deviation (\( s \)) is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
In our context, the standard deviation of the differences, \( s_{\text{diff}} \) = 7.2 feet, expresses the variability in height differences of the trees from 2009 to 2019.
Knowing the standard deviation is vital for calculating the margin of error and the confidence interval as it reflects how much the sample mean could vary from the true population mean. This measure helps gauge how uncertain or certain the estimate is.