A psychologist measures reaction times (in seconds) for a simple cognitive task in a sample of $n=50$ participants. The sample standard deviation of reaction times is $s=0.15$ s. Assuming reaction times are (approximately) normally distributed, construct a $95\%$ confidence interval (up to two decimal places) for the true population standard deviation $\sigma$.

A quality‐control engineer is interested in the variability of caffeine concentration (in $\text{mg/mL}$) in a particular brand of espresso. A simple random sample of $n=30$ cups is taken, and the sample standard deviation is found to be $s=0.35\text{ mg/mL}$. Assuming the population of concentrations is approximately normal, using a $95\%$ confidence level, determine the degrees of freedom, the critical values ${\chi }_L^2$ and ${\chi }_R^2$, and the confidence interval estimate (up to two decimal places) for the population standard deviation $\sigma$.

A food scientist wishes to estimate the variability in fiber content (in grams per serving) of a new high‑fiber breakfast cereal. A simple random sample of $n=36$ servings yields a sample standard deviation of $s=1.20$g. Assuming fiber content is normally distributed, using a $99\%$ confidence level, determine the degrees of freedom, the critical values ${\chi }_L^2$ and ${\chi }_R^2$, and the confidence interval estimate (up to two decimal places) for the population standard deviation $\sigma$.

A biologist records the weights in grams of two species of birds.

Species X$:15.2,15.5,15.1,15.3,15.4$

Species Y$:14.8,15.0,15.2,14.9,15.1$

Construct a $95$ percent confidence interval estimate for the variance of each species' weights. Is there evidence of a difference in variation between the two species?

A quality control team collects a random sample of $30$ batteries and finds that the sample variance in battery life is $11.56$ hours². Assume the data are from a normally distributed population. Construct the $95\%$ confidence interval for the population variance and the population standard deviation.