A precision instrument is checked by making 12 readings on...

A precision instrument is checked by making 12 readings on the same quantity. The population distribution of readings is normal. a. The probability is 0.95 that the sample variance is more than what percentage of the population variance? b. The probability is 0.90 that the sample variance is more than what percentage of the population variance? c. Determine any pair of appropriate numbers, $a$ and $b$, to complete the following sentence: The probability is 0.95 that the sample variance is between $a \%$ and $b \%$ of the population variance.

A precision instrument is checked by making 12 readings on the same quantity. The population distribution of readings is normal.
a. The probability is 0.95 that the sample variance is more than what percentage of the population variance?
b. The probability is 0.90 that the sample variance is more than what percentage of the population variance?
c. Determine any pair of appropriate numbers, $a$ and $b$, to complete the following sentence: The probability is 0.95 that the sample variance is between $a \%$ and $b \%$ of the population variance.

Key Concepts

Sample Variance

The sample variance is a measure of the dispersion of a set of measurements around their mean. When computed from a sample, it is used as an unbiased estimator of the population variance. However, its probability distribution is not normal but is related to the chi-square distribution, which allows for probabilistic statements linking the sample variance and the true population variance.

Confidence Intervals for Variance

Confidence intervals for a population variance are constructed by using the chi-square distribution quantiles corresponding to the chosen confidence level. The interval is determined by inverting the chi-square distribution so that the sample variance, when scaled appropriately by (n-1)/population variance, falls between the lower and upper quantile values, giving a range within which the true variance is expected to lie with a specified probability.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available to estimate a parameter. In variance estimation from n observations, using the mean as an estimated parameter reduces the available independent information to n-1 degrees. This adjustment is crucial for correctly applying the chi-square distribution in statistical inference about the variance.

Chi-Square Distribution

In the context of variance estimation, when sampling from a normally distributed population, the statistic formed by multiplying the sample variance by (n-1) and dividing by the population variance follows a chi-square distribution with n-1 degrees of freedom. This property is fundamental in constructing confidence intervals for the population variance and determining probabilities associated with observed sample variances.

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