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The sample mean, often denoted as \( \bar{x} \), is the average value of a set of observations and a basic measure of central tendency. Calculating it is straightforward: simply sum up all observations and divide by the number of observations. For the data given in the exercise, where elephant birth weights were 133, 120, 97, 106, and 124 kilograms, the sample mean is calculated as:

• Add the weights: 133 + 120 + 97 + 106 + 124 = 580 kilograms
• Divide by the number of elephants, which is 5: \( \bar{x} = \frac{580}{5} = 116 \) kilograms

The sample mean provides a center point of the data, allowing us to make inferences about the population from which the sample is drawn. It's essential for constructing the confidence interval, as it is the midpoint around which the interval is calculated.

Sample standard deviation, denoted as \( s \), is a measure of the amount of variation or dispersion in a set of values. Essentially, it indicates how much individual data points deviate from the sample mean.
To calculate the sample standard deviation for our elephant data, use the following steps:

• Calculate the variance, which is the average squared deviation from the sample mean: \( s^2 = \frac{(133 - 116)^2 + (120 - 116)^2 + (97 - 116)^2 + (106 - 116)^2 + (124 - 116)^2}{4} \)
• Compute each squared deviation: \[ (133 - 116)^2 = 289, (120 - 116)^2 = 16, (97 - 116)^2 = 361, (106 - 116)^2 = 100, (124 - 116)^2 = 64 \]
• The sum of squared deviations: 289 + 16 + 361 + 100 + 64 = 830
• The variance: \( \frac{830}{4} \approx 207.5 \)
• The standard deviation is the square root of the variance: \( s = \sqrt{207.5} \approx 15.59 \) kilograms

This measure helps determine the reliability and precision of the sample mean, affecting the width of the confidence interval.

The Central Limit Theorem (CLT) is a fundamental concept in statistics, crucial when constructing confidence intervals. It states that the sampling distribution of the sample mean will tend to be normal or bell-shaped if the sample size is sufficiently large, about 30 or more, even if the population distribution is not normal.
In the context of the exercise, the relatively small sample size of 5 means we rely on the assumption that elephant birth weights are approximately normally distributed. If this assumption is true, the means of different samples would create a normal distribution, leading to more accurate predictions about the population mean.

• For larger sample sizes, the shape of the collecting mean distribution would be approximately normal, irrespective of the population distribution.
• The CLT allows researchers to make conclusive statements using probabilities and statistics, even with non-normal data, as long as the sample size is sufficiently large or the population is normal.
• In small samples like our case, the CLT's reliance on normality is crucial for the validity of the confidence interval.

Understanding the CLT helps reinforce why certain statistical methods work, especially when estimating measures about the population with limited data.