Angels1234
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what is the correct definition for a confidence interval , there is specific wording so i dont want to get it wrong and guess. Also when calculating the confidence intervals do you have to give a full answer or is the standard 2/3 sf ok ?
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Shaanv
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(Original post by Angels1234)
what is the correct definition for a confidence interval , there is specific wording so i dont want to get it wrong and guess. Also when calculating the confidence intervals do you have to give a full answer or is the standard 2/3 sf ok ?
2/3 sf is good.

As for a definition go with the one in textbook or spec.

The confidence interval is a region in which there is a certain percentage chance of finding the population parameter.

95% confidence interval- 95% chance that population parameter (almost certainly the mean in S3) lies in this interval.

Thats the jist of it, have a look in ur textbook or on spec for a more concrete definition, or better yet see if u can find a past paper question asking for definition.
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Angels1234
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(Original post by Shaanv)
2/3 sf is good.

As for a definition go with the one in textbook or spec.

The confidence interval is a region in which there is a certain percentage chance of finding the population mean.

95% confidence interval- 95% chance that population mean lies in this interval.

Thats the jist of it, have a look in ur textbook or on spec for a more concrete definition, or better yet see if u can find a past paper question asking for definition.
thanks for the great answers as per
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Shaanv
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(Original post by Angels1234)
thanks for the great answers as per
Ive amended my post, have a quick look at the change.
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Gregorius
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(Original post by Shaanv)
95% confidence interval- 95% chance that population parameter (almost certainly the mean in S3) lies in this interval.
You have to be very careful with the language used about confidence intervals! As stated, the above is meaningless, as the population parameter is not a random variable, so you cannot make probability statements about it! The correct statement is that there is a 95% chance that the confidence interval encloses the population parameter. Best to check, as ever when exams are concerned, the textbook definition.
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Angels1234
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(Original post by Gregorius)
You have to be very careful with the language used about confidence intervals! As stated, the above is meaningless, as the population parameter is not a random variable, so you cannot make probability statements about it! The correct statement is that there is a 95% chance that the confidence interval encloses the population parameter. Best to check, as ever when exams are concerned, the textbook definition.
But doesn’t this mean that there’s a 95 percent chance that the interval contains mean . I read something about how the confidence level is NOT about there being a 95% chance that the mean is in the interval . What’s the difference between the 2 variations I’ve mentioned and why is the latter wrong?
Thanks
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Gregorius
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(Original post by Angels1234)
But doesn’t this mean that there’s a 95 percent chance that the interval contains mean . I read something about how the confidence level is NOT about there being a 95% chance that the mean is in the interval . What’s the difference between the 2 variations I’ve mentioned and why is the latter wrong?
Thanks
Let's take this from first principles. You start off with a population (for example, the height of all men in the UK aged between 18 and 35), which you believe can be described with reasonable accuracy by a handful of parameters (for example, normally distributed with a certain mean and standard deviation). You are interested in estimating what these population parameters are, and you are also interested in understanding how precise your estimates will be. Let's fix attention on estimating the population mean.

In order to make your estimate of the population mean, you draw a random sample and calculate the mean of this sample. This is your estimate of the population mean. You also calculate a 95% confidence interval (which provides your estimate of the precision with which the sample mean estimates the population mean). You hope that the sample mean is close to the population mean and you hope that your confidence interval contains the population mean.

But you don't have to stop there! Draw another random sample and calculate a confidence interval corresponding to this random sample. Do it again and again and again. You end up with a series of confidence intervals, C1, C2, ..., Cn, where n is a big number.

If you've calculated your confidence intervals correctly, and the assumptions you've made about the underlying population are correct, then, as n gets larger and larger, the expected number of 95% confidence intervals that actually contain the population mean will n x 0.95.

This then is the probability statement you can make about confidence intervals - any single confidence interval has a 95% chance of enclosing the true population mean. Notice that the population mean is a fixed, non-random quantity; you cannot make probability statements about it.
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Angels1234
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(Original post by Gregorius)
Let's take this from first principles. You start off with a population (for example, the height of all men in the UK aged between 18 and 35), which you believe can be described with reasonable accuracy by a handful of parameters (for example, normally distributed with a certain mean and standard deviation). You are interested in estimating what these population parameters are, and you are also interested in understanding how precise your estimates will be. Let's fix attention on estimating the population mean.

In order to make your estimate of the population mean, you draw a random sample and calculate the mean of this sample. This is your estimate of the population mean. You also calculate a 95% confidence interval (which provides your estimate of the precision with which the sample mean estimates the population mean). You hope that the sample mean is close to the population mean and you hope that your confidence interval contains the population mean.

But you don't have to stop there! Draw another random sample and calculate a confidence interval corresponding to this random sample. Do it again and again and again. You end up with a series of confidence intervals, C1, C2, ..., Cn, where n is a big number.

If you've calculated your confidence intervals correctly, and the assumptions you've made about the underlying population are correct, then, as n gets larger and larger, the expected number of 95% confidence intervals that actually contain the population mean will n x 0.95.

This then is the probability statement you can make about confidence intervals - any single confidence interval has a 95% chance of enclosing the true population mean. Notice that the population mean is a fixed, non-random quantity; you cannot make probability statements about it.
heya, thanks for the very detailed reply

I dont seem to understand this bit .."
You also calculate a 95% confidence interval (which provides your estimate of the precision with which the sample mean estimates the population mean). "
From what i understand for the rest on the info , are you saying we take different samples and for each of them calculate a confidence interval (where you choose whether you want a 90/95/99 confidence interval and each of the confidence intervals we calculate have all got to be for 90 or all for 99 confience right). So are we basically calculating confidence intervals for each individual sample? Also in a question eg in a book or an exam question are we trying to work out the confidence intervals for the entire population right .. based of all the confidence intervals from our indivual samples ? or are you saying we know that for each confidence interval there is a 95 percent chance that the population mean is contained within that interval. Also how does n x 0.95 link to exam questions and textbook questions or did you just mention this to give me a bit of background theory?
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Gregorius
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(Original post by Angels1234)
heya, thanks for the very detailed reply

I dont seem to understand this bit .."
You also calculate a 95% confidence interval (which provides your estimate of the precision with which the sample mean estimates the population mean). "
A 95% confidence interval has a 95% probability of enclosing the true sample parameter. The confidence interval also contains your estimate of the population parameter. So a narrow confidence interval means that your estimate of the population parameter has a high probability of being close to the actual population parameter.

From what i understand for the rest on the info , are you saying we take different samples and for each of them calculate a confidence interval (where you choose whether you want a 90/95/99 confidence interval and each of the confidence intervals we calculate have all got to be for 90 or all for 99 confience right). So are we basically calculating confidence intervals for each individual sample?
What actually happens in practice is that we have one sample, from which we calculate our estimate of the population parameter, and a confidence interval for that estimate.

Now, at the level you are studying, probability has a frequentist interpretation - if an event has a 95% probability of happening in a trial, you mean that if the trial is performed repeatedly, the event will happen in 95% of the trials. So my explanation was to provide you with the correct frequentist interpretation of a confidence interval - if you sample over and over, 95% of the confidence intervals would cover the actual population parameter.

Also in a question eg in a book or an exam question are we trying to work out the confidence intervals for the entire population right .. based of all the confidence intervals from our indivual samples ?
Nope. You work out a single confidence interval for your estimate of a population parameter. In all the questions you'll be given, you only get one sample! My repeated sampling illustration is intended to help you understand how a confidence interval is to be interpreted.

or are you saying we know that for each confidence interval there is a 95 percent chance that the population mean is contained within that interval.
Nearly, if you were able to sample repeatedly and obtain many confidence intervals. But you have to be careful about language here: "there is a 95% chance that the confidence interval encloses the population mean". NOT "there is a 95 percent chance that the population mean is contained within that interval". The reason for this is that the population mean is not random; you can't make probability statements about it.

Also how does n x 0.95 link to exam questions and textbook questions or did you just mention this to give me a bit of background theory?
Background theory!
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Angels1234
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(Original post by Gregorius)
A 95% confidence interval has a 95% probability of enclosing the true sample parameter. The confidence interval also contains your estimate of the population parameter. So a narrow confidence interval means that your estimate of the population parameter has a high probability of being close to the actual population parameter.



What actually happens in practice is that we have one sample, from which we calculate our estimate of the population parameter, and a confidence interval for that estimate.

Now, at the level you are studying, probability has a frequentist interpretation - if an event has a 95% probability of happening in a trial, you mean that if the trial is performed repeatedly, the event will happen in 95% of the trials. So my explanation was to provide you with the correct frequentist interpretation of a confidence interval - if you sample over and over, 95% of the confidence intervals would cover the actual population parameter.



Nope. You work out a single confidence interval for your estimate of a population parameter. In all the questions you'll be given, you only get one sample! My repeated sampling illustration is intended to help you understand how a confidence interval is to be interpreted.



Nearly, if you were able to sample repeatedly and obtain many confidence intervals. But you have to be careful about language here: "there is a 95% chance that the confidence interval encloses the population mean". NOT "there is a 95 percent chance that the population mean is contained within that interval". The reason for this is that the population mean is not random; you can't make probability statements about it.



Background theory!

estimates the population mean). " …see more
A 95% confidence interval has a 95% probability of enclosing the true sample parameter. The confidence interval also contains your estimate of the population parameter. So a narrow confidence interval means that your estimate of the population parameter has a high probability of being close to the actual population parameter.

Why is it the sample parameter and not the population parameter in the first sentance? Also for the 2nd sentance why does the CI contain my estimate of the population parameter, why is it my estimate , why isnt isnt it that the confidence interval contains the true population parameter Mu /population mean?? For the third sentance is the confidence interval has a smaller width doesnt that means there is less confidence so a smaller probability of the population parameter being contained within the interval .


thanks again
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Gregorius
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(Original post by Angels1234)

Why is it the sample parameter and not the population parameter in the first sentance?
Agh! Typo, my apologies. It should read "population parameter". Samples don't have parameters!

Also for the 2nd sentance why does the CI contain my estimate of the population parameter, why is it my estimate , why isnt isnt it that the confidence interval contains the true population parameter Mu /population mean??
Confidence intervals are constructed by design (sample mean plus or minus two standard deviations, for example) to enclose the estimate of the population parameter. As an individual sample could (by random chance) be quite weird, it is possible that an confidence interval will not contain the population parameter. In fact. for a 95% confidence interval, you would expect a confidence interval to miss the population parameter 5% of the time!

For the third sentance is the confidence interval has a smaller width doesnt that means there is less confidence so a smaller probability of the population parameter being contained within the interval .
No, the probability that the confidence interval contains the population parameter is fixed (at 95%, say). If the confidence interval contains the population parameter (which it will 95% of the time) and it contains your estimate (which it does by design), then a narrow confidence interval forces the estimate to be near to the population parameter.
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