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When dealing with sample data, understanding the variance helps identify how much the values deviate from the mean. Sample variance is denoted as \( s^2 \). It's calculated using the formula: \( s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2 \), where \( x_i \) are the sample points, \( \bar{x} \) is the sample mean, and \( n \) is the number of samples. This formula helps adjust for bias, making it a better estimate of the population variance than directly calculating the variance of the data set. To calculate it, you:

• Find the mean of your sample data.
• Subtract the mean from each data point and square the result.
• Sum all the squared differences.
• Divide by the number of data points minus one.

In the example given with Lysine levels, the sample variance was found to be \( 3.13 \) after performing these steps, highlighting the spread in the sampled Lysine composition levels in soybean meal.

The Chi-square distribution is a key concept in statistics, especially when dealing with variance and constructing confidence intervals. It is represented by \( \chi^2 \) and is essential because it helps in making inferences about the population variance from sample variance. The Chi-square distribution with \( n-1 \) degrees of freedom is used when estimating variance. The degrees of freedom here refer to the number of values in the final calculation of a statistic that are free to vary. Important to note, the distribution is not symmetric, and its shape depends heavily on the degrees of freedom. In the example, a 99% confidence interval for the sample variance requires finding \( \chi^2 \) values from the Chi-square table for specific alpha levels. This yields interval bounds that reflect potential true population variance values, given the sample.

Lower confidence bounds give a minimum possible value for a parameter with a certain confidence level. When calculating a lower confidence bound for variance \( \sigma^2 \), it shows the smallest value that could be expected, considering sample variability.To calculate the lower bound, the formula \( \frac{(n-1)s^2}{\chi^2_{\alpha}} \) is used, where \( \chi^2_{\alpha} \) is determined by alpha, the level of significance. It provides insight into the minimum magnitude of variance, protecting against overestimating certainty.For the Lysine data, a 99% lower confidence bound for \( \sigma^2 \) is calculated and found to be noticeably higher at \( 13.49 \), which indicates the degree of variability in soybean Lysine levels that one can confidently describe as at least this large.

A two-sided confidence interval estimates a range in which a parameter like variance \( \sigma^2 \) lies, with a certain level of confidence, usually expressed as a percentage.In statistical terms, it means that if the same sampling method were repeated numerous times, the true parameter would lie within these bounds a certain percent of the time (e.g., 99%).To construct a two-sided confidence interval for variance, we use \( \left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \right) \). Finding these values involves using the appropriate \( \chi^2 \) distribution for the given confidence level.The given exercise calculated this interval for Lysine levels to be (1.30, 10.44), providing a broader picture of the variability in soybean meal that could be expected with high confidence.