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The concept of the **Sample Mean** is fundamental in statistics. It represents the average value of a sample and is used to estimate the mean of the entire population. Calculating the sample mean involves a straightforward process, especially when you break it down step by step:

• Add all the sample measurements together.
• Divide the total by the number of observations in the sample.

In our exercise, copper content measurements were provided: 66, 66, 65, 64, 66, 67, 64, 65, 63, and 64. The formula looks like this:\[\bar{x} = \frac{66 + 66 + 65 + 64 + 66 + 67 + 64 + 65 + 63 + 64}{10} = 65\]The **Sample Mean**, denoted as \( \bar{x} \), is 65. This value provides a summary measure of the central location of your data points.
It acts as a "balancing point" for the data set, simplifying comparisons among different samples.

Calculating the **Sample Standard Deviation** gives us an understanding of how spread out the data is around the mean. It reflects the variability or dispersion of the sample data.Here's how you can calculate it using a series of steps:

• Find the deviation of each sample point from the sample mean.
• Square each deviation to avoid negatives affecting the sum.
• Sum these squared deviations.
• Divide by \( n-1 \), where \( n \) is the number of observations, to find the average squared deviation.
• Take the square root of this result to get the standard deviation.

In our scenario, the sample standard deviation for the copper content is approximately calculated as follows:\[s = \sqrt{\frac{1}{9} \sum (x_i - \bar{x})^2}\]After performing the calculation, \( s \approx 1.4907 \).
This value indicates a fairly tight clustering of the data around the mean, suggesting that the data points do not deviate much from the average.

The **t-Distribution** is used when making inferences about a sample mean and when the sample size is small (usually \( n < 30 \)) or the population standard deviation is unknown. It is a type of probability distribution that is similar to the standard normal distribution but has heavier tails.Here are some of its characteristics:

• Symmetrical about the mean.
• Has a mean of zero.
• The tails are heavier compared to standard normal distribution, accounting for more variability.

For our confidence interval calculation, the **t-Distribution** provides a critical value, \( t \), which is used to calculate the margin of error. This value depends on:

• Desired confidence level (e.g., 99%).
• Degrees of freedom, which is \( n-1 \) (in our case, 9).

Consulting a **t-Distribution** table or calculator, the critical value for \( 99\% \) confidence and \( 9 \) degrees of freedom is approximately \( 3.249 \).
This value inflates the interval to account for the uncertainty inherent in estimating the population mean with a small sample.