Suppose that you have obtained data by taking a random sample from a population and that you intend to find a confidence interval for the population m

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 - ?${\displaystyle <p<}$?

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 j: denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence intervals (7-8, 9.6)
(a) Would 2 90%% confidence interval calculated from this same sample have been narrower or wider than the glven interval? Explain your reasoning.
(b) Consider the following statement: There is 9 95% chance that Is between 7.8 and 9.6. Is this statement correct? Why or why not?
(c) Consider the following statement: We can be highly confident that 95% of al bottles ofthis type of cough syrup have an alcohol content that is between 7.8 and 9.6. Is this statement correct? Why or why not?

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before​ treatment, 13 subjects had a mean wake time of 101.0 min. After​ treatment, the 13 subjects had a mean wake time of 94.6 min and a standard deviation of 24.9 min. Assume that the 13 sample values appear to be from a normally distributed population and construct a ${\displaystyle 95\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with drug treatments. What does the result suggest about the mean wake time of 101.0 min before the​ treatment? Does the drug appear to be​ effective?
Construct the ${\displaystyle 95\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with the treatment.
$min<\mu <min$ ​(Round to one decimal place as​ needed.)
What does the result suggest about the mean wake time of 101.0 min before the​ treatment? Does the drug appear to be​ effective?
The confidence interval ▼ does not include| includes the mean wake time of 101.0 min before the​ treatment, so the means before and after the treatment ▼ could be the same |are different. This result suggests that the drug treatment ▼ does not have | has a significant effect.

A random sample of 100 automobile owners in the state of Virginia shows that an automobile is driven on average 23,500 kilometers per year with a standard deviation of 3900 kilometers.
Assume the distribution of measurements to be approximately normal.
a) Construct a ${\displaystyle 99\mathrm{\%}}$ confidence interval for the average number of kilometers an automobile is driven annually in Virginia.
b) What can we assert with ${\displaystyle 99\mathrm{\%}}$ confidence about the possible size of our error if we estimate the average number of kilometers driven by car owners in Virginia to be 23,500 kilometers per year?

Maximum error in confidence interval
A sample of 40 cows is drawn to estimate the mean weight of a large herd of cattle. If the standard deviation of the sample is 96 kg, what is the maximum error in a 90% confidence interval estimate?

A poll asked the question, "What do you think is the most important problem facing this country today?" Twenty-five percent of the respondents answered "crime and violence." The margin of sampling error was plus or minus 3 percentage points. Following the convention that the margin of error is based on a ${\displaystyle 95\mathrm{\%}}$ confidence interval, find a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the percentage of the population that would respond "crime and violence" to the question asked by the pollsters.
Lower limit ${\displaystyle \mathrm{\%}}$
Upper limit ${\displaystyle \mathrm{\%}}$

The bad debt ratio for a financial institution is defined to be the dollar value of loans defaulted divided by the total dollar value of all loans made. A random sample of 6 Ontario banks is selected and that the bad debt ratios (in percentages) for these banks are:
${\displaystyle 7,4,6,7,5,8}$
Compute and interpret a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean bad debt ratio. What needs to be true in order for this interval and interpretation to be valid?