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A student was asked to find a 90% confidence interval for the proportion of students who take notes using data from a random sample of size n 75. Which of the follo...

Question

A student was asked to find a 90% confidence interval for the proportion of students who take notes using data from a random sample of size n 75. Which of the following is a correct interpretation of the interval 0.14 0.352 Check all that are correctThere is a 90% chance that the proportion of the population is between 0.14 and 0.35. The proprtion of all students who take notes is between 0.14 and 0.35, 90% of the time_ There is a 90% chance that the proportion of notetakers in a sample of 75 st

A student was asked to find a 90% confidence interval for the proportion of students who take notes using data from a random sample of size n 75. Which of the following is a correct interpretation of the interval 0.14 0.352 Check all that are correct There is a 90% chance that the proportion of the population is between 0.14 and 0.35. The proprtion of all students who take notes is between 0.14 and 0.35, 90% of the time_ There is a 90% chance that the proportion of notetakers in a sample of 75 students will be between 0.14 and 0.35_ With 90% confidence, the proportion of all students who take notes is between 0.14 and 0.35 With 90% confidence, a randomly selected student takes notes in a proportion of their classes that is between 0.14 and 0.35_



Answers

Explaining confidence The admissions director from Big City University found that $(107.8,116.2)$ is a
95$\%$ confidence interval for the mean 1$Q$ score of all freshmen. Comment on whether or not each of the following explanations is correct.
(a) There is a 95$\%$ probability that the interval from 107.8 to 116.2 contains $\mu .$
(b) There is a 95$\%$ chance that the interval $(107.8,$ 116.2 ) contains $\overline{x}$ .
(c) This interval was constructed using a method that produces intervals that capture the true mean in 95$\%$ of all possible samples.
(d) 9$\%$ of all possible samples will contain the interval $(107.8,116.2) .$
(e) The probability that the interval $(107.8,116.2)$ captures $\mu$ is either 0 or $1,$ but we don't know which.

In party. I want to find peak cap. Okay, So what will be peak app? My sample sizes. 5000 and 717 people returned. So I peek at is 717 upon 5000. This is my proportion. My sample proportion. If I use a calculator, this is going to be seven months. Seven divided by 5000 0.14340 point 1434 How do I calculate the margin of error? My margin of error is Z Alfa by two, multiplied by route over Big Gap Que cap by. And the end is the sample size, which is 5000. My Q is equal to my cue cap is going to be one minus p cap. So if I do that, I get 0.85660 point 8566 Okay, what is my Alfa By two? I want a 90% confidence. So my Alfa is 0.1 and my Alfa by two will be 0.50 point zero Fine. This is my Alfa by two. So if I use it calculated to calculate the critical value this is 1.6449 So I just put in the values 1.6449 multiplied by root off 0.1434 multiplied by 0.8566 upon 5000. So after using the calculator, let's just do that. This is going to be rude off 0.1434 multiplied by 0.85 66 divided by faith closing. So this is 0.49 fight. And if I multiply this with one point 6449 I get this a 0.815 This is this Value turns out to be 0.8150 point 00815 The third parties calculate the confidence interval. How do we do that? We know confidence. Interval is peak ap minus e less than peak less than p cap. Plus, if we know that our true proportion with lie between these two values. So what does this come out to be? It is 0.1350 point 135 less than people less than 0.152 0.152 Okay, now, how do it interpret this? We are 90% confident that this interval off 0.135 to 0.15 to actually does contain the true value of the population. Proportion off returned surveys. So this is going to be my answer.

The internal given between 25 30 is a 90% confidence interval estimate for the population means you we want to interpret what this finding suggests. That is What does it mean to say that an interval is a 90% confidence interval for a mean? You so remember that confidence intervals are going to estimate a population stat with some percent confidence. However, that percent confidence does not allude to whether or not new is in the individual interval after all new is either not in or in a confidence interval with probability one or zero. It's either there or it's not there. So the question is then, what is the confidence intervals confidence level actually estimating? So we know that the 90% then must allude to the fact that 90% of intervals built by these samples. So by samples of similar size are going to contain you. So where are percent confidence is whether or not this general type of sample for the given sample size Will contain you. That means we can interpret this interval to mean if 90 of confidence intervals are calculated from all possible samples of a given size me will be a 90 of the confidence intervals.

So we know that first proportion is .18 And the second proportion is .16. And we want to find, so that's are different. So at .2 difference Plus remind us. And then we want the team of disease star value for 90 confidence, that's the 1.645. And then we find our big standard deviation. And for the confidence interval you do not use that waited pool, you use these actual values. So I take the .18 Times 2.82 Divided by the sample size which was 10,000. And I take the .16 Times its complement .84 divided by the sample size. That was 8000. And then we'll take this point oh two minus this quantity and the 20.2 plus that quantity. And when we do we get the interval comes out to be 0.10 But 10076, Our .01076 UP 2.0- 9- four. So we think that the difference between those actual proportions Is sandwich somewhere in here with 90 confidence

The following is a solution to # seven. And this says that a random sample of 12 people were selected. We don't really know what it is, but it's from a normal approximation or a normal distribution approximately normal. So that allows us to have this small sample size. Normally, you have to have a sample size of at least 30. But since it comes from a normal distribution, we can just say that this is okay, so in equals 12 is fine. And the uh the sample mean X bar was 45 And the Sample Standard Deviations 14. And we're asked to find the 90% confidence interval for the population mean. So, since we're estimating the population mean, we're either going to use the Z or the tea interval, and we're gonna use the tea interval, because we don't know what sigma is, we don't know that population standard deviation, because we weren't given given that we were only given the sample standard deviation. S so because of that, we have to use the T interval. Okay, so terrible. So the 90% confidence interval now you can use the tea interval formula, you can certainly go about it that way. I'm gonna use the form of the calculator just because it goes by a little bit more quickly. But if you go to stat and then tests, it's gonna be this eighth option down here, this is on a. T. I. T. For uh T. Interval, so I'm gonna click eight and make sure summary stats is Is highlighted there and I'm just gonna start filling in my stuff. So x bar is 45. The sample standard deviation is 14 And the n. was 12 And we want to be 90%. So I want to say .9 there for the sea level And we calculate and then this top band here gives us what we need. So 37 742 and 52.258. Let's go and write that down. So 37 742 All the way up to 52 258 It doesn't say to do this, but if it asks for an interpretation, you would just say we are 90% confident that the true population mean mu is between 37.742 and 52.258.


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