Assume the sample is from a normally distributed population...

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for $(a)$ the population variance $\sigma^{2}$ and $(b)$ the population standard deviation $\sigma .$ Interpret the results. The average annual precipitations (in inches) of a random sample of 30 years in San Francisco, California, have a sample standard deviation of 8.18 inches. Use a $98 \%$ level of confidence.

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for $(a)$ the population variance $\sigma^{2}$ and $(b)$ the population standard deviation $\sigma .$ Interpret the results.
The average annual precipitations (in inches) of a random sample of 30 years in San Francisco, California, have a sample standard deviation of 8.18 inches. Use a $98 \%$ level of confidence.

Key Concepts

Confidence Interval for Variance

A confidence interval for the variance is a range of values, calculated from sample data, that is likely to contain the true population variance with a specified level of confidence. When the underlying distribution is normal, the chi-square distribution is used to derive the interval, applying the formula that involves dividing the product of the sample variance and degrees of freedom by chi-square quantiles.

Confidence Interval for Standard Deviation

A confidence interval for the standard deviation is determined by taking the square root of the endpoints of the confidence interval for the variance. This interval provides a range within which the true population standard deviation is expected to lie with a certain level of confidence, reflecting uncertainty associated with sample estimates.

Chi-square Distribution

The chi-square distribution plays a critical role in the inference about variance and standard deviation when sampling from a normal population. It governs the distribution of the scaled sum of squared deviations from the mean, and its quantiles are used to construct confidence intervals for these parameters.

Degrees of Freedom

Degrees of freedom, typically defined as the sample size minus one, represent the number of independent values that can vary in the calculation of a statistic. In the context of variance and standard deviation estimation, they adjust the chi-square distribution which in turn refines the construction of confidence intervals.

Interpretation of Confidence Intervals

Interpreting a confidence interval involves understanding that it is a range derived from sample data which, under repeated sampling, will contain the true parameter (such as variance or standard deviation) a specified proportion of the time. It reflects the uncertainty in the estimate rather than implying a probability about the parameter itself.

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