Master Confidence Intervals: Examples, Equations, and Questions

A certain reported that in a survey of 2006 American adults, ${\displaystyle 24\mathrm{\%}}$ said they believed in astrology.
a) Calculate a confidence interval at the ${\displaystyle 99\mathrm{\%}}$ confidence level for the proportion of all adult Americans who believe in astrology. (Answers should be rounded to three decimal places.)
(_______, _______)
b) What sample size would be required for the width of a ${\displaystyle 99\mathrm{\%}}$ CI to be at most 0.05 irresoective of the value of ${\displaystyle \hat{p}}$? (Round your answer up to the nearest integer.)

It was observed that 30 of the 100 randomly selected students smoked. Find the confidence interval estimate of ${\displaystyle 95\mathrm{\%}}$ confidence level for ${\displaystyle \pi }$ ratio of smokers in the population.
a) ${\displaystyle P\left(0.45<p<0.78\right)=0.95}$
b) ${\displaystyle P\left(0.11<p<0.28\right)=0.95}$
c) ${\displaystyle P\left(0.51<p<0.78\right)=0.95}$
d) ${\displaystyle P\left(0.21<p<0.59\right)=0.95}$
e) ${\displaystyle P\left(0.21<p<0.39\right)=0.95}$

$\begin{array}{|ccccc|}\hline 25.6& 23.8& 26.5& 23.9& 24.8\\ 24.7& 20.9& 26& 22.2& 27\\ \hline\end{array}$
What Equation to use to construct a ${\displaystyle 99\mathrm{\%}}$ lower confidence bound for ${\displaystyle {\sigma }^2}$

Express the confidence interval ${\displaystyle 53.5\mathrm{\%}<p<69.7\mathrm{\%}}$ in the form of $\hat{p}\pm ME$

The average annual mileage of a car is 23,500 kilometers, with a standard deviation of 3900 kilometers, according to a sample of 100 Virginians who own cars.
Assume that the measurement distribution is roughly normal.
a) Create a 99 percent confidence interval for the typical Virginian vehicle's annual mileage in kilometers.
b) If we assume that Virginians drive an average of 23,500 kilometers per year, what can we say with 99 percent certainty about the potential extent of our error?

A survey of several 10 to 11 year olds recorded the following amounts spent on a trip to the mall:
$19.17,$21.18,$20.38,$25.08
Construct the ${\displaystyle 90\mathrm{\%}}$ confidence interval for the average amount spent by 10 to 11 year olds on a trip to the mall. Assume the population is approximately normal.
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Step 4 of 4 : Construct the ${\displaystyle 90\mathrm{\%}}$ confidence interval. Round your answer to two decimal places.

The following data, recorded in days, represents the recovery time, for patients who are randomly treated with one of two medications to cure servere bladder infections:
$\begin{array}{|cc|}\hline Medication1& Medication2\\ n_1=13& n_2=16\\ {\overline{x}}_1=20& {\overline{x}}_2=15\\ s_1^2=2.0& s_2^2=1.8\\ \hline\end{array}$
Find the ${\displaystyle 99\mathrm{\%}}$ confidence interval for ${\displaystyle \mu 1-\mu 2}$, the difference in mean drug recovery times, and INTERPRET it to get a helpful conclusion about the drugs.
Assume normal populations, with equal variances.