The quality-control manager of a chemical company randomly...

The quality-control manager of a chemical company randomly sampled twenty 100 -pound bags of fertilizer to estimate the variance in the pounds of impurities. The sample variance was found to be 6.62 . Find a $95 \%$ confidence interval for the population variance in the pounds of impurities.

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available for estimating a parameter. When estimating variance from a sample of size n, the degrees of freedom are typically n-1, which directly affects the critical values from the chi-square distribution used in constructing the confidence interval.

Confidence Interval for Variance

A confidence interval for variance provides a range of plausible values for the true population variance based on the observed sample variance. It is constructed by scaling the sample variance with quantiles from the chi-square distribution corresponding to a specified level of confidence.

Chi-square Distribution

The chi-square distribution is a probability distribution that arises when a set of independent, standard normal random variables is squared and summed. It is crucial in constructing confidence intervals for variance because the sampling distribution of (n-1)*s^2/?^2 follows a chi-square distribution with n-1 degrees of freedom.

Random Sampling and Estimation

Random sampling is the process of selecting a subset of observations from a larger population in such a way that each individual has an equal chance of being included. This method ensures that the sample is representative of the population, thereby supporting unbiased estimation of parameters like the population variance.

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