In a Gallup poll of $450$ employed adults, $82\%$ said they prefer working remotely at least part of the time. Among the $450$ respondents, how many said they prefer working remotely?

A news website asks visitors to vote online on whether a newly released public service advertisement is “respectful” or “disrespectful.” Of the $1500$ visitors who responded, $1230$ said the ad was respectful and $270$ said it was disrespectful. Construct a $95\%$ confidence interval estimate of the true proportion of all viewers who believe the ad is respectful.

A technician is testing the consistency of a machine that cuts sheets of cardboard. The sample variance of the thickness of $20$ cardboard sheets is found to be $25{\text{ mm}}^2$. Assume the thickness values are normally distributed. Construct a $90\%$ confidence interval for both the population variance and the population standard deviation.

A survey is conducted among $3500$ high school principals in a country where their yearly bonuses are known to have a non-normal, right-skewed distribution. Suppose many random samples of $35$ principals are taken, and the average yearly bonus is calculated for each sample. What value do these sample means aim to estimate?

A standardized test consists of multiple-choice questions, each with four answer choices, only one of which is correct. Suppose you randomly guess on three such questions, and the correct answer for each is "b". What is the mean of the sampling distribution of the sample proportion of correct answers?

A poll indicates that $59\%$ of employees at a company are satisfied with their work-life balance. If $80$ employees are randomly chosen, what is the probability that the sample proportion satisfied is less than $50\%$?

Why are degrees of freedom important in a Chi Square Test for Independence?

Find the critical values ${\chi }_R^2$ and ${\chi }_L^2$ for a confidence level of $c=0.90$ with a sample size of $n=21$.

A sample of waiting times in minutes at a local pharmacy is $3,6,5,4,7,5,6,4,5,6$. Construct a $95\%$ confidence interval 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 pediatrician measures head circumferences (in units of cm) of $12$ healthy newborns and finds a sample standard deviation of $s=0.49$ cm. To gauge the precision of this estimate, she takes $10$ bootstrap samples (with the replacement) of size $12$ and computes each sample’s standard deviation. The sorted bootstrap standard deviations (in cm) are: $0.45,0.48,0.49,0.50,0.51,0.52,0.54,0.56,0.58,0.60$. Using these $10$ values, construct a $95\%$ bootstrap confidence interval for the true population standard deviation $\sigma$.