A sample of $40$ students has a mean test score of $78$ with a standard deviation of $10$. Construct a $95\%$ confidence interval for the population mean.

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 sport psychologist measures the change in reaction time (in units of seconds) for $12$ athletes before and after a specialized training program. Negative values indicate a faster reaction time post‑training. The observed sample is: $\{\text{–}0.4,0.1,\text{–}0.2,0.3,0.0,\text{–}0.1,0.5,\text{–}0.3,0.2,0.4,\text{–}0.5,0.6\}$. Ten bootstrap samples (with the replacement) of size $12$ are drawn, and their means (in ascending order) turn out to be: $\text{–}0.47,\text{–}0.32,\text{–}0.18,\text{–}0.05,0.08,0.17,0.29,0.41,0.52,0.67$. Using only these ten bootstrap means, construct an $80\%$ confidence interval estimate for the true mean change in reaction time.

A city planner wants to estimate the mean commute time in minutes for residents. The population standard deviation is $15$ minutes. What is the minimum sample size needed to be $90\%$ confident that the sample mean is within $4$ minutes of the true mean?

A pharmaceutical company wants to estimate the mean weight of pills produced by a new machine within $0.12$ grams. If the population standard deviation is known to be $0.5$ grams, what is the minimum sample size required to construct a $99\%$ confidence interval for the population mean?

A researcher collects a simple random sample of waiting times in minutes for a popular amusement park ride. A confidence level of $95\%$ was used. The researcher used a calculator for analysis, as shown in the figure.

Provide a statement that correctly interprets the confidence interval displayed.

A university reports that the average score on its entrance exam is $72$ with a standard deviation of $8$. If random samples of $64$ students are selected, what are the mean and standard deviation of the sampling distribution of the sample mean?

The annual salaries (in units of dollars) for $10$ randomly chosen dental hygienists are listed below. Assume the population is normally distributed.

Construct a $95\%$ confidence interval for the population mean annual salary for dental hygienists. Round your answer to the nearest whole number.

A weather balloon records the following pairs of altitude in thousands of feet and temperature in ${}^{\circ }F$: $(1.5,60)$, $(3.8,52)$, $(6.0,44)$, $(8.2,35)$, and $(10.5,28)$. Find (a) the explained variation, (b) the unexplained variation.

What is the critical value $t_c$ for a $95\%$ confidence level when the sample size is $21$?