Master Statistics and Probability Problems with Expert Help

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

In a random sample of 24 people, the mean commute time to work was 33.7 minutes and the standard deviation was 7.1 minutes. What is the margin of error of ${\displaystyle \mu }$?
Interpret the results.
A) With ${\displaystyle 90\mathrm{\%}}$ confidence, it can be said that the population mean commute time as between the bounds of the confidence interval.
B) It can be said that ${\displaystyle 90\mathrm{\%}}$ of people have a commute time is between the bounds of the confidense interval
C) With ${\displaystyle 90\mathrm{\%}}$ confidence, it can be said that the commute time is between the bounds of the confidence interval.
D) If a large sample of people are taken approximately ${\displaystyle 90\mathrm{\%}}$ of them will have commute times between the bounds of the confidence interval.

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. (Round your answers 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.)

The Level of Significance of a Statistical Test The ${\displaystyle R}$ and $D$ department of a paint company has developed an additive that it hopes will increase the ability of the company’s stain for outdoor decks to resist water absorption. The current formulation of the stain has a mean absorption rate of 35 units. Before changing the stain, a study was designed to evaluate whether the mean absorption rate of the stain with the additive was decreased from the current rate of 35 units. The stain with the additive was applied to 50 pieces of decking material. The resulting data were summarized to ${\displaystyle \bar{x}=33.6}$ and ${\displaystyle s=9.2}$
a) Is there substantial evidence ${\displaystyle (\alpha =0.01)}$ that the additive reduces the mean absorption from its current value?
b) What is the level of significance (p -value) of your test results?
c) What is the probability of a Type II error if the stain with the additive in fact has a mean absorption rate of 30?
d) Estimate the mean absorption using a 99% confidence interval. Is the confidence interval consistent with your conclusions from the test of hypotheses?

A standard normal table, also called the unit normal table or Z table, may be a mathematical table for the values of ${\displaystyle \varphi }$, which are the values of the cumulative distribution function of the traditional distribution.
$\begin{array}{|cc|}\hline \text{confidence intervals}& Z\\ 90& 1.645\\ 95& 1.96\\ 96& 2.05\\ 97& 2.17\\ 98& 2.326\\ 99& 2.576\\ \hline\end{array}$
a) How do calculate Z-score using the confidence interval?

In a survey of 2695 adults, 1446 say they have started paying bills electronically in the last year.
Construct a ${\displaystyle 99\mathrm{\%}}$ confidence interval for the population proportion. 
Interpret your results. Choose the correct answer below. 
A) With ${\displaystyle 95\mathrm{\%}}$ confidence, it can be said that the sample proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval 
B) With ${\displaystyle 99\mathrm{\%}}$ confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval. 
C) The endpoints of the given confidence interval show that adults pay bills online ${\displaystyle 99\mathrm{\%}}$ of the time.

A set of data consists of 45 observations between 0 and 29. Which size would you advise for the class period?

A random sample of 820 households from a city, Stats City, were surveyed. From the statistical technology, we find a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean number of people in the households was ${\displaystyle 1.85<\mu <3.25}$
What is the true meaning of the interval presented in the problem about Stats City?

Suppose that the height, in inches, of a 25-year-old man is a normal random variable with parameters mu=71 and sigma^2=6.25 . What percentage of 25-year-old men are over 6 feet, 2 inches tall? What percentage of men in the 6-footer club are over 6 feet, 5 inches?

You are interested in finding a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean number of visits for physical therapy patients. The data below show the number of visits for 14 randomly selected physical therapy patients. Round answers to 3 decimal places where possible.
$\begin{array}{|cccccccccccccc|}\hline 9& 6& 10& 15& 19& 6& 23& 26& 19& 16& 11& 25& 16& 11\\ \hline\end{array}$
a) To compute the confidence interval use a t or z distribution.
b) With ${\displaystyle 95\mathrm{\%}}$ confidence the population mean number of visits per physical therapy patient is between "?" and "?" visits.
c) If many groups of 14 randomly selected physical therapy patients are studied, then a different confidence interval would be produced from each group. About "?" percent of these confidence intervals will contain the true population mean number of visits per patient and about "?" percent will not contain the true population mean number of visits per patient.