A fitness center wants to estimate, with $90\%$ confidence, the proportion of its members who attend workout sessions at least three times a week. The estimate must be accurate within $3\%$ of the true population proportion. A previous survey showed that $83\%$ of members attend workout sessions this frequently. What is the minimum sample size needed?

In a poll of $950$ college students, $55\%$ reported that they prefer online classes to in-person classes. Construct a $95\%$ confidence interval for the proportion of all college students who prefer online classes. Does the interval suggest that more than half of all college students prefer online classes?

If the sample size increases while the confidence level remains the same, what happens to the margin of error in a confidence interval for a population proportion?

Two surveyors, Aisha and Ben, are tasked with estimating the confidence interval for the proportion of electric cars in a city. They both start with the same point estimate, which is $0.09$. Aisha calculated her interval to be from $0.07$ to $0.11$, while Ben's interval is from $0.08$ to $0.12$. Which surveyor's confidence interval is incorrect?

A confidence interval for the proportion of customers satisfied with a new product is given as $(0.702,0.766)$. Determine the sample proportion and the margin of error.

In a study, group A had $x_1=132$ successes out of $n_1=600$, and group B had $x_2=95$ successes out of $n_2=480$. Construct a $98\%$ confidence interval for $p_1-p_2$.

The number of defective light bulbs found in $10$ randomly selected shipments had a sample standard deviation of $4.2$ bulbs. Assume the number of defects follows a normal distribution. Using an $80\%$ confidence level, construct a confidence interval for the population standard deviation (${\sigma }^{}$). Also, interpret the results.

A streaming service company plans to introduce a new premium tier and wants to estimate the worldwide percentage of its users who intend to subscribe. Based on a pilot, $20\%$ of users indicated they plan to upgrade. The marketing team needs a $90\%$ confidence interval with a margin of error of $3$ percentage points. How many users must be surveyed? Can the team simply poll their own friends and colleagues instead of drawing a random sample? Explain.

A researcher wants to estimate the difference in proportions of two populations. In sample $1$, $x_1=52$ successes out of $n_1=220$. In sample $2$, $x_2=41$ successes out of $n_2=180$. Construct a $90\%$ confidence interval for $p_1-p_2$.

A marketing manager for a streaming service wants to estimate the percentage of adults in the U.S. who subscribe to a streaming platform. How many adults must be surveyed to be $99\%$ confident that the estimate is within $2$ percentage points of the true population percentage? Assume that $65\%$ of all adults currently have a streaming service subscription, based on recent industry reports.