Confidence Interval for Population Means in Statistics

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“Welcome to this lesson and mastering statistics. We re going to continue talking about the the confidence interval for a couple of simpler problems. We re not going to get the full glory here on how to calculate these confidence intervals. Just yet i want to solidify some more fundamental concepts with them first so these problems will be fairly simple in retrospect.

But they ll be important for you to build your knowledge and understanding. So here. We have the first problem. A survey of 200 males shows that they read on average of fifteen point seven hours per.

Week if the margin of error is 22. Hours at a 95 confidence level construct the confidence interval all right so there s two key things that you need to understand before you can solve this problem. The first thing is we discussed previously that when we take a survey like that or a sample like that and get the sample mean that the point estimate for the population. Mean is going to be equal to the sample mean so here we re sampling 200 males showing how often they read or how much they read fifteen point seven hours a week.

We re trying to find a confidence interval that confidence interval is going to tell us how many hours per week. The population will read at and it s going to be about you know a lower bound and an upper bound that will contain the population means. What we re trying to do so let s write things down. We know from the problem that the sample mean is equal to fifteen point seven hours that is what we have survey two hundred males right and we also know the margin of error e.

Which said we would denote that with e is two point two hours. So this is hours and hours and we need to calculate a confidence interval. Very very simple to do really once you understand and remember that the mean. Plus.

The margin of error is fifteen point seven plus two point two will give you 17 point nine. And the mean minus the margin of error is fifteen point seven minus two point two and you get thirteen point five. Now if you remember from previous discussions..

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We said that whenever we get for the sample mean we re going to take that as a point for our population. Mean 157. So in other words. We surveyed these 200 people we re going to assume since the survey result resulted in an answer of fifteen point seven hours.

We re going to assume that the center of our confidence interval is at that at that point. We re going to assume as a point estimate that the population mean of a whole population is fifteen point seven. We know that s not right that s why we re given the margin of error. And so this confidence.

Interval is going to extend up from the point estimate to seventeen point nine. And it s also going to extend below that guy to thirteen point five. So do you see how the point estimate lies in the center of this thing that we re calling the confidence interval. So the way you write it down.

So way you would write it down is you would say 95 confidence interval whoops. Enter goal is the following the way you would write it down is you would say that the population mean is less than seventeen point. Nine and greater than thirteen point five the way you read this for those of you who haven t done a math class. In a while.

When you have two inequalities like that and you have the the thing of interest in the middle. This is the population mean this is this is everybody in the population. The population is probably going to be males in whatever country or city or you re talking about and the parameter you re looking at is how many hours per week. That they read so we re saying the population mean is going to fall between these limits here.

So you read it from the inside out you go here. And you read this way the mean is greater than thirteen point five. The mean is less than seventeen point nine..

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So you can write it that way another way you could write it in terms of interval notation is just put the lower number thirteen point five comma and then the upper number okay. I like representing it like this because i like seeing that the actual variable there. But this is perfectly fine to you re late you re listing a confidence interval so this whole answer is really it of course. You can choose to display it either way.

Here. Now why is it a 95 confidence interval well. We haven t really gotten into the details of that yet in this. Problem you were told that the margin of error was 22.

In hours at 95 confidence so we re bypassing. Some details and future problems. I ll give you the tools to be able to calculate all that yourself here you ve been given the margin of error. And you need to construct the interval so for your purposes.

You just subtract the margin of error and add the margin of error just to make it absolutely explicit the distance from the point estimate to the lower guy is the margin of error. The distance from the center to the upper part is also the margin of error. So you go from the point estimate down one margin of error and up one margin of error that s how you construct the confidence interval. So similar type of problem we ll just do it for.

Practice a survey of 600 people finds that they sleep an average of 105 hours per night if a margin of error at. 98 confidence interval is 13 hours a 13. Hours construct the confidence interval. So we re given the how many people we.

Sampled we re told what the sample mean is 105 hours in this case. We re given the level of confidence. 98..

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And we re given the error margin of error. Which is one point three hours. So really it s the same sort of. Thing the sample mean after we do the surveying is.

25 hours the margin of error is 13. Hours and in this particular problem. It doesn t really matter. The confidence level or the number of people we surveyed that s just kind of extra information in future problems.

You will use that information to calculate the confidence interval. But here you ve given the margin of error. So all you really have to do is say ok well ten point five plus one point three is eleven point eight and ten point five minus one point three is nine point two you go. Plus one margin of error in minus of margin of error.

And so to write that down you can say we are 98 confidence interval that the population mean is going to be greater than 92. And less than eleven point eight or if you want to write it more as an interval opening parenthesis and say nine point two comma. Eleven point eight either. One is a acceptable.

So what this is saying here to interpret the results. What you re saying is you ve constructed an interval that ranges from nine point two to eleven point eight in this case. And this is the number of hours per night. That you get sleep right and we re saying that the population average is going to fall between these numbers and we re 98 certain 98.

Certain that if we could actually talk to everybody in the country and get an answer from everybody and average. It together that the average number of hours per week that we get from that is going to fall between these boundaries here so you see it s really impossible to do that if we could just get answers from everybody then we wouldn t have to ever use really statistics too much so. What we do then is we sample a portion of people we get an answer and then we calculate this confidence interval and say well we we don t know everything..

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But we re pretty darn sure how sure in this case. We re 98. Sure that the population mean is going to fall in this interval. That s what the confidence interval is and that s why it s important.

It s very very useful especially when you re looking at manufacturing. You know you might say you re 98 percent. Confident that your defect rate is going to fall within a certain range there however many defects per week or whatever. But you can t study every single cell phone coming off the line.

So. What you do is you sample maybe 500 of them. And get a good number based on that now these two problems have been pretty simplified because i ve been given the sample mean i ve been given the margin of error right now in future problems you re going to learn how to calculate this margin of error yourself. And that margin of error is going to be dependent on the level of confidence that you have there.

And it s also going to be dependent upon your sample size. So we haven t done any of that in these problems. I m just kind of crawling before we walk once you get that margin of error. Though all you do is you add it and you subtract.

It to get the confidence and let you have so follow me on to the next section. Where we ll build these skills you ll continue learning about confidence intervals and how to calculate them and we ll do it one step at a time and i think you ll see this topic is one ” ..

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