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The concept of the sample mean is foundational in statistics. The sample mean is essentially the average of a selected set of data points. Here, it represents the average annual consumption of alcohol by young European women based on the given sample. To compute the sample mean, we sum all the observations in the dataset and then divide by the number of observations.Here's a quick refresher on the formula:

• Sum all data points: \( \sum{x_i} \)
• Count the number of data points: \( n \)
• Calculate the mean: \( \bar{x} = \frac{\sum{x_i}}{n} \)

In the provided example, the sum of all values is 3870, and the number of observations is 20. Taking these into account, we have a sample mean \( \bar{x} \) of 193.5 liters. This sample mean is a key estimate of the true population mean and is used throughout the confidence interval calculation.

Standard deviation is a measure of how spread out numbers are in a dataset. It provides insights into the variability or dispersion of the sample, helping us understand how much data points typically deviate from the mean. A lower standard deviation indicates that the data points tend to be close to the mean, while a higher standard deviation suggests more spread.To calculate the standard deviation in a sample:

• Subtract the mean from each data point to find the deviation \( (x_i - \bar{x}) \)
• Square each deviation to eliminate negative values and sum them up: \( \sum{(x_i - \bar{x})^2} \)
• Divide by the number of observations minus one (\( n - 1 \)), which accounts for the degrees of freedom in a sample: \( \frac{\sum{(x_i - \bar{x})^2}}{n-1} \)
• Take the square root to return the value to the original unit: \( s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}} \)

For the provided dataset, this calculation results in a standard deviation of 291.51 liters. Knowing the standard deviation is crucial for determining the confidence interval.

The t-distribution is a probability distribution that is used when the population standard deviation is unknown and the sample size is small. It's similar to the normal distribution but has heavier tails, which means there's a higher probability for extreme values. As the sample size increases, the t-distribution approaches the normal distribution.In the context of the confidence interval:

• We use the t-distribution because our sample size is less than 30, which is a general threshold to switch from using the z-distribution (normal) to the t-distribution.
• The shape of the t-distribution depends on the degrees of freedom, which is \( n - 1 \). In this case, \( n-1 = 19 \).
• The t-value corresponding to a 95% confidence level for 19 degrees of freedom is approximately 2.093.

This t-value is crucial for computing the margin of error, as it adjusts the interval based on the sample size and variability. Understanding the t-distribution helps in making more accurate estimates of a population parameter using a sample.

The margin of error provides a range that accounts for the uncertainty inherent in sampling. It defines how much the sample mean could potentially differ from the true population mean. A larger margin of error indicates greater uncertainty, while a smaller one represents more confidence in the sample mean.To calculate the margin of error in the context of a confidence interval:

• Find the t-value for your confidence level and degrees of freedom. Here, it's 2.093 for a 95% confidence level with 19 degrees of freedom.
• Use the formula: \( ME = t \cdot \frac{s}{\sqrt{n}} \) where:
  • \( t \) is the t-value
  • \( s \) is the standard deviation (291.51 liters in this case)
  • \( n \) is the number of observations (20 here)
• Substitute the known values to calculate ME: \( 2.093 \times \frac{291.51}{\sqrt{20}} = 136.46 \)

This margin of error, 136.46 liters, is added and subtracted from the sample mean (193.5 liters) to find the confidence interval. The final confidence interval informs us that the true mean consumption of alcohol likely falls between 57.04 and 329.96 liters for this sample.