Final Answer:
Degrees of freedom, in the context of the sample variance formula, ensure that (B) estimating the mean uses up some of the information in the data, and (C) are (n-1) when replacing the population mean by the sample mean.
Explanation:
In statistical analysis, degrees of freedom (df) play a crucial role in determining the precision of estimates. In the context of the sample variance formula, they represent the number of independent pieces of information available after estimating certain parameters. When calculating sample variance, the mean of the sample is estimated, using up some information from the data. For instance, in the sample variance formula \(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\), the mean \(\bar{x}\) is estimated, and this estimation uses up one degree of freedom.
Furthermore, regarding the replacement of the population mean by the sample mean, degrees of freedom are (n-1) in the formula for sample variance, where 'n' is the number of observations. When using the sample mean in place of the population mean, it is essentially an estimation from the data itself, which reduces the amount of independent information available by one. Therefore, to adjust for this loss of information, the degrees of freedom are (n-1) in the formula, ensuring an unbiased estimate of the population variance based on the sample.
In summary, degrees of freedom in the context of sample variance signify the loss of information due to estimating parameters from the sample data. They ensure the accuracy of estimates by accounting for the reduction in independent information caused by estimating the sample mean and are crucial in determining the precision and reliability of statistical inferences based on sample data.
Therefore the best option for the answer is B) & C).