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CI CL for a simple porportion

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  • #50651

    newbie
    Participant

    Hi Forum,
    I want to determine the CI and CL for a proportion.  If a sample from a given process was taken and a yield measure extracted (500 trials / 51 events), can I determine the CI and CL of said yield? Or is this not possible with attribute data?
    If the CI isnt possible, could I use a one-proportion test to determine a CL…..”95% confident the yield is greater than /less than /equal to 89%”…..
    Thanks!!

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    #174417

    Bower Chiel
    Participant

    Hi Newbie
    Have a look at : -http://www.stat.tamu.edu/~jhardin/applets/signed/case6.html
    Good luck
    Bower Chiel

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    #174418

    newbie
    Participant

    Hey BC,
    Thanks but I couldnt open the link…I suppose I should go to the TX A&M stat department web page and look for….???
    Thanks!

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    #174421

    Darth
    Participant

    Yes, you can do a CI using either the binomial or the normal approximation.  For your example of 51 events from 500 trials the 95%CIs are:
    .075-.129 for the Z approximation and .083-.127 using the binomial
    I have a nifty confidence interval calculator I put together that does CIs for a bunch of difference situations.
     

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    #174424

    newbie
    Participant

    Hey Darth,
    Thanks for the feedback!  So:

    How do I “read” the 51/500 = .89 with a CI of (.075-.129)?

    95% confident the parameter lies between ??
    How do I use the CI here relative to the proportion?
    Can you steer me toward reference or online calculator so I can see how the computation is done?
    Thanks for your patience Darth!

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    #174434

    Darth
    Participant

    A CI is intepreted as “based on the sample, 95 out of 100 times a sample proportion will fall between .075 and .129”.  Sometimes you will also see it written as “Based on my sample, I am 95% confident that the true population proportion will fall between .075 and .129.”
    Catch me offline at [email protected] and possibly I can share my calculator.  You won’t find a comparable one online.

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    #174440

    DaveS
    Participant

    Drath,
    Want to get into a semantics fight? Your statement,
    “based on the sample, 95 out of 100 times a sample proportion will fall between .075 and .129”
    is not quite correct.
    I like the second statement much better.
    It really doesn’t matter for most purposes, thought you might enjoy thinking about this.

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    #174446

    Darth
    Participant

    Daves,
    While you might enjoy the simpler and more vernacular definition of a confidence interval, it is not technically correct although, as you say, for the layman such as yourself, it should suffice.  Below please find four references defining a confidence interval which I believe you will find support my original definition.  I tired and figured four should be enough for you to consider.  Possibly you learned something today.
    For a given proportion p (where p is the confidence level), a confidence interval for a population parameter is an interval that is calculated from a random sample of an underlying population such that, if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same method, a proportion p of the confidence intervals would contain the population parameter in question.
    If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter. Confidence intervals are usually calculated so that this percentage is 95%, but we can produce 90%, 99%, 99.9% (or whatever) confidence intervals for the unknown parameter.
     Confidence intervals are constructed at a confidence level, such as 95%, selected by the user. What does this mean? It means that if the same population is sampled on numerous occasions and interval estimates are made on each occasion, the resulting intervals would bracket the true population parameter in approximately 95% of the cases.
    Confidence intervals are written with a percentage; what does this percentage represent?   If the researcher were to take 100 random samples with a 95% confidence interval for each sample, then he or she expects that for 95 of the 100 samples (95%), the range of values produced by the confidence interval procedure will include the true mean of the population. 

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    #174450

    Michael Mead
    Participant

    Sorry Darth, I must agree with Daves (and you, in your second definition). When you say “95 out of 100 times” that sounds like an absolute number. What if I did repeated trial and I found that 100 of 100 sample means were within your margin of error? Would you be wrong?

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    #174452

    Darth
    Participant

    Possibly you should re read all the definitions. Are they all wrong? At a 95% confidence level it means that 95 out of 100 times. It also means that over time not for a few samples. There is such a thing as sampling error and variation. 95% also means 950 out of 1000 and 9500 out of 10000. Just as with the central limit theorem all conditions are predicated upon larger and larger samples starting to approach the hypothetical. A sample of 10 averages will likely not equal the population average. Does that mean the CLT is wrong? No, it means that it approaches as the number of samples gets larger and larger. For the CI, if you just do it 100 times using the many online simulators sometimes you get 95 sometimes 94 sometimes 97. That’s error not incorrectness. Do it 10000 times and it should now approach the 95% of the time.

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    #174454

    Michael Mead
    Participant

    Hello Darth,
    Both Daves and I understand the concept. The problem is putting an exact number on your estimate. One of the reasons we use confidence intervals is because we don’t want to be tied to a specific number, which might not be exactly right.  Your idea is correct, but your statement is not.  Thus, 95 % confidence is not the same as 95 out of 100 times.
    Once I had a QC supervisor who had a hard time understanding the stanrd normal distribution table. Her standard answer was “9 out of 10 times”. Of course with acceptance sampling her sampling plan was “so many, out of so many.”  Statistics is not precise so our definitions should be.

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    #174455

    DaveS
    Participant

    So you still stand by your original statement? You are wrong. 95 out of 100 samples will NOT of necessity be in any other CI constructed. By your own post you do state the true meaning that the population parameter will be in 95 0f 100 such intervals constructed. You exhibit an arrogance which I did not think you had. I was hoping you would reread your post and realize your mistake. By the way Dr. Darth. Do you have a statistics degree?You taught me nothing today other than the existence of your overwhelming ego.

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    #174457

    Gourishankar
    Participant

    Refer “statistics for management” by Levin & Rubin , 7th edition
    ….” we are 95% confident that the mean battery life of the population lies within 30 & 42 months”
    This statement does not mean that the chance is 0.95 that the mean life of all batteries falls within the interval established from this sample . Instead,  it means that if we select many random samples of the same size and calculate interval for each of these samples , then in about 95% of these cases , the population mean will lie within that interval.
    Gourishankar

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    #174467

    Darth
    Participant

    Thanks Gourishankar, that’s what I was trying to explain to that idiot Daves with two attempted posts and four explanations from various sources but his mind is weak. I used the 95 out of a hundred times as a simple example and tried to explain that you need to get a whole bunch of samples and that in the long run I would expect about 95% of those samples to fall within that range. That might require 1000 samples or 10000. He got hung up on the 95 out of 100. Your statement is a nice clarification. I also note that you say “about 95% of those cases”. The more samples we get the more stable that 95% gets and the less variation occurs.

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    #174472

    Mikel
    Member

    You are surprised by Darth’s arrogance? Wow, you must not be paying
    attention – he treats most dullards that way, get accustomed to it.

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    #174477

    Darth
    Participant

    Stan, glad to see ya back.  Guess you got your bag limit of lobsters this week so you have returned to civilization.  Hope all is well with the family and your sister in law has left the bum and is now available again.

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    #174478

    DaveS
    Participant

    Darth,You apparently still don’t get it. I am not at all concerned about the 95 out of 100, but rather your misstatement of the meaning of a confidence interval. For a confidence interval constructed such as the one the original poster posed, about 95 times out of 100 the population parameter of interest will be contained in that interval. Your 4 sources all say that. However, your original statement “based on the sample, 95 out of 100 times a sample proportion will fall between .075 and .129” is simply not true. Any other sample proportion taken may or may not be within the 0.075 to 0.129 interval BUT if we construct CIs around each sample taken in the same way, 95 out of 100 will contain the population parameter of interest. To try and make this crystal clear to you, suppose the true population proportion is 0.08. This is in the sample CI constructed by the original poster. Another sample may show a proportion of 0.06. This is NOT in the OPs first sample CI, BUT the CI constructed with the second sample will contain the 0.08 population parameter of interest. A third sample may yield a proportion of 0.13, again not in the original interval, but its CI will also contain the 0.08 parameter of interest. If we take 100 samples, there is no way to predict how many sample proportions will fall within the first interval constructed as you say, but we can say that on the long term average 95 out of 100 (or whatever confidence we assign e.g. 90,99 etc.) Cist from those samples constructed in the same way will contain the population parameter. That is why we can say that with 95% confidence we believe the population parameter is in the interval. We can not rigorously say (although it often is said) that there is a .95 probability that any one interval contains the population parameter because once the sample is taken and the CI constructed, that interval has either a 0 or 1 probability of containing the parameter. I don’t have it with me, but in “Statistical Intervals: A Guide for Practitioners”, Hahn and Meeker, I remember them making a statement to the effect that the confidence implies more a confidence on the method of forming the interval. You really ought to get that reference and read through their explanation.

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    #174482

    Taylor
    Participant

    UH Daves, its a homework question, he’s not taking multiple samples, he just has one sample
    Darth 1
    Daves 0

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    #174483

    DaveS
    Participant

    Chad,
    Who cares.
    The meaning of the confidence interval is the issue. You should learn to read.

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    #174484

    Darth
    Participant

    Daves,
    Before this exchange gets out of hand, let’s forget what has been said and see if we can agree on a definition.  A random sample is drawn from a population.  A 95% confidence interval is calculated from the sample average, s.d. and size.  What this means is that as we take more and more samples we expect that eventually 95% of subsequent samples should encompass the true population mean.  You OK with this?  This is a bit different than just saying that based on the one sample you are 95% sure that the true population falls between the computed values of the interval.

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    #174485

    Muffin’s Dad
    Participant

    Really a poorly articulated contrasting of Bayesian credible intervals with confidence intervals.   When you are deriving intervals comprised of the true values of parameters with a stated degree of confidence you are deriving credible intervals.  On your other likely somewhat misshapen hand, if you are truly deriving and using confidence levels you are not deriving inference about true values of parameters.   Inferences can be incorrect when confidence levels are misinterpreted as credible intervals – and especially when the credible levels are misthought of as confidence levels.   What an amazing load of crapola you’re floating into the Six Sigma channel of commerce.
     
    And you call yourself a physicist!  

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    #174486

    Taylor
    Participant

    Ok Daves, I’ve tried to be nice and even through some humor to the situation. I have read the post, and You just almost got there, so try reading this.
    CI is bounded by a Lower Limit and Upper Limit that are determined by the risk associated with making the wrong conclusion about the parameter of interest. FOR EXAMPLE if the 95% (95 out of 100 samples) or any other sample n, and the lower confidence limit and the upper confidence limit are determined to be X1 and X2, respectively, it can be stated with 95% CERTAINTY that the true population average lies between the values X1 and X2.
    the 95% CI could also show that 95 of 100 subgroups collected with the sample size n would contain the true population average.
    If another 100 subgroups were collected ninety five of the subgroups’ averages would fall within the upper and lower confidence limits
    Whether the true population average lies within the upper and lower confidence limits that were calculated cannot be known.
    Larger sample sizes result in tighter confidence intervals, as expected from the central limit theorem.
    Regards

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    #174487

    Darth
    Participant

    Vinny you certainly haven’t lost your way with verbose, never ending sentences using big words with confusing meanings.  But, that’s why we all love you.  And what’s with the physicist comment if you were referring to me.  You know I ain’t no stinkin physicist although some have tagged me as a psychotic.
    Thought of you the other night….get your mind out of  the gutter.  Darth Dog had chewed up some stuffed animal so all there was left was the outer fabric of the critter.  I picked it up off the floor and put it on Mrs. Darth’s head and declared that she is now the proud owner of one of Vinny’s original animal hair adornments.  Funny at the time for me but she was disgusted that I put that old chewed rag on her head.  I spent the night in the Death Star on the couch.

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    #174489

    DaveS
    Participant

    Darth, You said, “Before this exchange gets out of hand, let’s forget what has been said and see if we can agree on a definition. A random sample is drawn from a population. A 95% confidence interval is calculated from the sample average, s.d. and size. What this means is that as we take more and more samples we expect that eventually 95% of subsequent samples should encompass the true population mean. You OK with this?” I’m down with that. What I have said all along. “This is a bit different than just saying that based on the one sample you are 95% sure that the true population falls between the computed values of the interval.”How is it different?It appears you have abandoned your position on 95% of samples all being within any one calculated interval? That was the statement that I can find no reference for. Despite your assertions that your previous references support that, they do not. If you still cling to what I believe a falsehood, please provide something to support it. Merely saying it over and over is hardly proof. One may call a cat a dog, but the damn thing still will not bark. I believe the best statement of the meaning is the one you provided earlier…For a given proportion p (where p is the confidence level), a confidence interval for a population parameter is an interval that is calculated from a random sample of an underlying population such that, if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same method, a proportion p of the confidence intervals would contain the population parameter in question.There is still the little matter of your insults and
    calumny against me. I see no hint of an retraction. But bygones may be bygones.I hope you have learned something today. I try to every day.

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    #174501

    Darth
    Participant

    Daves,
    I am glad that we have settled upon a viable definition. If you can’t see the difference between the two statements, I won’t try to point it out again.As to my original statement, perhaps it was a bit hasty and not well worded. That’s the problem with expedient responses and I agree it could have been worded better.As for your hurt feelings, it is apparent that this is your first contribution and real posting on this site. If you are that thin skinned, possibly you shouldn’t engage in further discussions. Sometimes manners will slip a bit and by comparison, my “insults” were quite minor.As for what I learned today, I will take a bit more time in my explanations so whiners like you don’t get so hot and bothered. By the way, the original poster and I communicated offline and he is quite pleased with the CI calculator I sent him so mission accomplished.

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    #174502

    DaveS
    Participant

    Darth,Poor wording is a hallmark of a weak mind.
    Idiots like you do that a lot.I’m glad you did get the poster the calculator.

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    #174503

    Muffin’s Dad
    Participant

    Thanks, Darth. I just thought I’d take a minute to pile on some
    additional illogical and lightly researched statistics’ish critique.
    Daves should not get all the credit there.I gave you the physicist plug to make you feel better about getting
    your PhD in qual/stats thingys and having been a trainee of the
    great man “the first and real Dr. D.” himself – that had to have
    been a rocky yet rewarding career launch. (Plus not everyone can
    get their doctorate in ops research, you shouldn’t necessarily feel
    inadequate [solely] because of that…)Thanks for asking, Muffin is dong quite well and in spite of some
    alarming decomposition, still wears well from comfort, security and
    stylistic perspectives.

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    #174504

    Darth
    Participant

    Gee Vinny/Muffin, how did I know it was you?????  Perhaps you can take young Daves under your paw and straighten him out.  As a less than prolific contributor to this Forum, he certainly has an attitude reminiscent of previous name calling children.  Possibly he can resurrect the old J O club from the past.  Possibly he is old Joe or Reigle coming back from the beyond.  Maybe you can get him a job at the new company doing Wipe On Wipe Off.

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    #174508

    Mikel
    Member

    Limited out by 10:15 on Wednesday, spent an hour trading out small
    ones for bigger ones and was on Duval by 1:00. Shot grouper and got
    another limit by 2:00 on Thursday. Back in Tampa drinking wine at
    Bueanes Friday evening. Life is good. All is well with the family and babe of a sister is pregnant and due on
    Oct. 11.How are things with you?

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    #174561

    Don Howie
    Participant

    Dr. Ken Fledman, is that you?
     
    I cannot believe this!  I sent you my CI calculator for you to evaluate, and here you are giving it away.
     
    Did you test it?  The results I got were 7.37% – 12.63%  + or – 2.63% at 95%
     
    Can you review please?
     
    Don H.

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