Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let

In a science fair​ project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left​ hand, and then she asked the therapists to identify the selected hand by placing their hand just under​ Emily's hand without seeing it and without touching it. Among 358 ​trials, the touch therapists were correct 172 times. Complete parts​ (a) through​ (d).
a) Given that Emily used a coin toss to select either her right hand or her left​ hand, what proportion of correct responses would be expected if the touch therapists made random​ guesses? ​(Type an integer or a decimal. Do not​ round.)
b) Using​ Emily's sample​ results, what is the best point estimate of the​ therapists' success​ rate? ​(Round to three decimal places as​ needed.)
c) Using​ Emily's sample​ results, construct a ${\displaystyle 90\mathrm{\%}}$ confidence interval estimate of the proportion of correct responses made by touch therapists.
Round to three decimal places as​ needed - ?${\displaystyle <p<}$?

In a study of the accuracy of fast food​ drive-through orders, Restaurant A had 298 accurate orders and 51 that were not accurate. a. Construct a 90​% confidence interval estimate of the percentage of orders that are not accurate. b. Compare the results from part​ (a) to this 90​% confidence interval for the percentage of orders that are not accurate at Restaurant​ B: ${\displaystyle 0.127<p<0.191}$. What do you​ conclude? a. Construct a 90​% confidence interval. Express the percentages in decimal form. ___

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.)

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let e be the level of confidence used to construct a confidence interval from sample data. Let αα be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters such as p, '${\mu }_1-{\mu }_2$, or $p_1-p_2$, which we will study in Section 9.3, 10.2, and 10.3.) Whenever the value of k given in the null hypothesis falls outside the c=1−α confidence interval for the parameter, we reject $H_0$. For example, consider a two-tailed hypothesis test with \alpha =0.01 and $H_0:\mu =20H_1:\mu 20$ sample mean x¯=22 from a population with standard deviation σ=4.
(a) What is the value of c=1−α. Using the methods, construct a 1−α confidence interval for μ from the sample data. What is the value of μ given in the null hypothesis (i.e., what is k)? Is this value in the confidence interval? Do we reject or fail to reject H0 based on this information?
(b) using methods, find the P-value for the hypothesis test. Do we reject or fail to reject $H_0$? Compare your result to that of part (a).