A researcher collects a simple random sample of size $n=12$ from a population that is normally distributed. The sample mean, $\bar{x}$, is $95$, and the sample standard deviation, $s$, is $8$. Construct a $98\%$ confidence interval for the population mean $\mu$. How does decreasing the sample size affect the margin of error, $E$?

A campus survey estimates the standard deviation of hours students study per week as $s=12.4$. How many students are needed to estimate the mean study hours within $3$ hours, with $95\%$ confidence?

A health study estimates the standard deviation of daily step counts as $s=9.8$. The minimum number of participants required to estimate the mean daily steps within $2$ steps, with $99\%$ confidence, is found to be $n=160$. What effect does doubling the required accuracy have on the sample size?

Population $X$ has a standard deviation ${\sigma }_X=8$, and population $Y$ has a standard deviation ${\sigma }_Y=12$. How many times larger than Population $X$'s sample size does Population $Y$'s sample size need to be to achieve the same margin of error for the mean?

A nutritionist wants to estimate the difference in the proportion of adults and teenagers who consume at least five servings of fruits and vegetables daily. What sample size should be obtained from each group if she wants the estimate to be within $4$ percentage points with $95\%$ confidence, using previous estimates of $28\%$ for adults and $22\%$ for teenagers?