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 $\sigma$.

Chicago Commute Times: The following are $15$ commute times in minutes for a random sample of Chicago residents: $10,35,20,15,60,10,30,45,10,55,20,10,25,40,10$. Construct a $99\%$ confidence interval for the population mean commute time. Is this interval a precise estimate of the mean?

A health organization wants to estimate the average cholesterol level of adults in a city. They require $99\%$ confidence that the sample mean is within $4$ milligrams per deciliter of the population mean. Prior data indicates that the population standard deviation is $22.5$ milligrams per deciliter. What is the minimum sample size that is needed? Would sampling only from one clinic be appropriate?

A researcher wants to estimate the mean weight of a certain species of fish. The population standard deviation is known to be $3.2$ grams. If the sample size is $64$ and the confidence level is $99\%$, what is the margin of error for this estimate?

A company claims that the average delivery time for its packages is $48$ hours with a standard deviation of $4$ hours. If a random sample of $32$ deliveries is selected, what is the probability that the sample mean delivery time is more than $47$ hours?

A confidence interval for the average test score in a class is given as $(72.4,79.6)$. What are the sample mean and the margin of error?

A study reports that the mean daily water consumption per household in a certain city is $245$ liters, with a standard deviation of $38$ liters. Suppose random samples of $64$ households are selected, and the mean daily water consumption for each sample is calculated. What are the mean and standard deviation of the sampling distribution of sample means?