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. A magazine includes a report on the prices of subcompact digital cameras. The article states that 11 randomly selected subcompact digital cameras have a sample standard deviation of $\$ 109 .$ Use an $80 \%$ 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.
A magazine includes a report on the prices of subcompact digital cameras. The article states that 11 randomly selected subcompact digital cameras have a sample standard deviation of $\$ 109 .$ Use an $80 \%$ level of confidence.

Key Concepts

Assumption of Normality

The assumption that the data come from a normally distributed population is crucial when constructing confidence intervals for variance and standard deviation using the chi-square distribution. This assumption ensures that the distribution of the statistic used in the interval estimation follows a chi-square distribution, thus allowing for accurate computation of the interval bounds.

Chi-square Distribution

The chi-square distribution plays a central role in constructing confidence intervals for variance and standard deviation when the underlying population is normal. It characterizes the distribution of the scaled sum of squared deviations and provides the critical values needed to set the bounds of the confidence interval.

Confidence Level

The confidence level, such as 80%, represents the probability that the method used to construct the confidence interval will capture the true population parameter if the same process is repeated on multiple independent samples. It reflects the degree of certainty or reliability of the estimation procedure.

Confidence Interval for Variance

A confidence interval for the variance provides a range of values within which the true population variance is expected to lie, based on the sample variance and a chosen confidence level. Constructing this interval involves using the chi-square distribution, which is appropriate when the data come from a normally distributed population.

Confidence Interval for Standard Deviation

A confidence interval for the standard deviation is derived by taking the square roots of the lower and upper bounds of the variance confidence interval. This interval gives an estimated range for the population standard deviation with a defined confidence level, effectively translating the uncertainty from variability to dispersion in the same units as the original data.

Please give Ace some feedback