A statistics practitioner took a random sample of 43 observations from a population whose standard deviation is 26 and computed the sample mean to be

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?

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

Consider the next 1000 98% Cis for mu that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of $\mu ?$
What isthe probability that between 970 and 990 of these intervals conta the corresponding value of ? (Hint: Let
Round your answer to four decimal places.)
‘the number among the 1000 intervals that contain What king of random variable s 2) (Use the normal approximation to the binomial distribution

A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is ${\displaystyle \sigma =15}$
a) Compute the ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population mean. Round your answers to one decimal place.
b) Assume that the same sample mean was obtained from a sample of 120 items. Provide a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population mean. Round your answers to two decimal places.
c) What is the effect of a larger sample size on the interval estimate?
Larger sample provides a-Select your answer-largersmallerItem 5 margin of error.

Use the Composite Trapezoidal, Simpson's and Midpoint Rules to approximate the integral
${\int }_1^{2}x\ln (x)dx,n=4$
(For the Midpoint Rule use $n+2$ sub intervals.)

Sample mean to population mean
How to use sample mean to know the population mean? Now I have sample mean, sd and population mean. They are 37.28, 25 and 34. Sample size is 25 Here is my thought ${\displaystyle N<25}$. Sample mean may not equal to population mean. population mean is ${\displaystyle 37.28+\frac{25}{\sqrt{25}}=42}$ Am i correct?

Sampling Distribution of the Sample Mean
Concerning estimations/intervals:
True or False: if the three conditions Random, Normal, and Independent for using a confidence interval for a population mean are not met, then the sampling distribution of ${\displaystyle \bar{x}}$ is unknown.
I'm unsure about the criteria - would you not know the distribution of the sample mean even if these criteria are unfilled? Are there specific conditions for doing so?