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Use of a t-table (critical value)

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  • #51934

    engineer
    Participant

    How can I find the “right” critical value? The following things are known:
    12 pieces are randomly selected. Their mean is 200. A hypothesis test is performed with null hypothesis Ho: mu = 200 and alternative hypothesis Ha: mu < 200. Alpha should be 0.01.
    I am trying to find out the right t-value. Because I have 12 values, the degrees of freedom should be 11. Now my question: It is a one tail test so that I need to look for Alpha (0,01; 11 degrees of freedom) = 2.718? That is what I think.
    However the solution should be another value: 3.106. That might mean I should  better look for Alpha/2 because it is a two-tail test? Then Alpha (0.005; 11 degrees of freedom)  would be 3.106.
    This is part of an exam preparation question but I am afraid it is rather poorly prepared because even I have discovered some errors. Could someone give me a hand – I am rather confused.
    Thanks!

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    #181825

    Ken Feldman
    Participant

    I think the first thing you want to do is provide a s.d.

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    #181827

    Obiwan
    Participant

    Darth…slight disagreement…the first thing you want to do is to get your hypotheses corrected.  The null and alternate that you proposed are not mutually exclusive.
    Obiwan

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    #181828

    Ken Feldman
    Participant

    Expound on what you mean.

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    #181829

    Obiwan
    Participant

    Darthy-Dear
    In the original post, the null was stated with an equals sign and the alternate with a less than symbol.  If you use the equals sign for the null, then the alternate MUST be not-equals.  If you use the less than symbol for the alternate, then the null MUST be greater than or equals.  Basic hypotheses…they must be mutually exclusive conditions…
    Obiwan
     
    P.s. but, of course, you knew that…

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    #181832

    Ken Feldman
    Participant

    Thanks for clarification Obi. Please explain if the Null was equal to 200 and the alternate was less than 200 why they are not mutually exclusive? There is no overlap on the values. Here is the output for an example. Doesn’t seem like Mini had a problem.One-Sample T: C1 Test of mu = 50 vs < 50
    95% Upper
    Variable N Mean StDev SE Mean Bound T P
    C1 100 52.00 10.10 1.01 53.68 1.98 0.975

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    #181838

    Obiwan
    Participant

    Darth
    In the situation where the null is equal to 200 and alternate is 200?  You have not created a situation where these are the hypotheses are the only available options…there is a third option that is not represented.
    Thinking about it, “mutually exclusive” is not the words I was searching for…the hypotheses need to represent all possible outcomes of the decision.
    Obiwan

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    #181841

    Craig
    Participant

    Engineer….your first inclination is correct: 
    It is a one tailed test because the alternate is mu < 200.
    Df is 11, and you find t-crtical under the alpha = .01 column in the table.
    If the alternate was mu not equal 200, you would divide alpha by two, and look in the .005 column.
     

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    #181850

    Ken Feldman
    Participant

    Obi,
    Guess I got hung up on your words. Let me try it this way. If the null is Ho=200 and the Ha200? I can’t simultaneously fail both the null and alternate. From what you are saying, we need an alternate that would cover the contingency of the truth being the mean >200. Right? In that case, you are saying the Ha needs to be not equal to which would cover both the less than and greater than condition. By looking at the resulting means we can tell whether it is larger or smaller in value assuming we reject the null. Right? So, the original post and the question he is trying to answer is poorly written, right?

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    #181857

    Obiwan
    Participant

    Darth…well said…
    Obi

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    #181863

    Reade
    Participant

    Darth and Obiwan,
    Your discussion about how to state the hypotheses isn’t incorrect.  However, in D. Montgomery’s Design and Analysis of Experiments, 7th ed., all hypotheses are stated in the style the original poster used.
    Hacl’s response is correct – since all possibilities are not included in H0 and H1, it is a 1-tailed test.
    Cathy

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    #181864

    Obiwan
    Participant

    Cathy
    Sorry, but I believe that I am correct.  I did not have ready access to that particular Montgomery book, but I did look into a couple of his other statistics books…and what you state is as he says it.  That does not make it correct.
    In a one tail test, you still have to cover all possibilities for the decisions.  Those possibilities are “equals,” “less than,” and “greater than.”  Thus, the null hypothesis in the case stated should be “greater than or equal to”…not just equals.  Why?
    The mathematics of the hypothesis test do not know the orientation of the data, thus the person applying the test must understand the orientation.  I created a test data set in Minitab, with the first set of 30 data points having a mean of 10 and sd of 1, with the second set of 30 having a mean of 20 and sd of 1.  Then I applied the test as you propose…when test set one is the first set tested, I get a p value of 0, thus we reject the null hypothesis. 
    When I reverse the test sets, I get a p-value of 1.0, which, when intrepreted says that the data sets are equal?  If I were actually using this data to manage a process, I would thus say that the two sets of data are equal…when we know that they are not.  If you include in the null hypothesis, the statement of “greater than or equal”…then you have covered all positions.
    I stand by my original posting, maybe modifying it a bit, that in hypothesis testing, you should always cover all potential outcomes.  And, while I am loathe to argue with Doug Montgomery, I believe that his books should be modified to cover all possible outcomes.
    Thanks…for making me look up some information and attempt to validate my point.
    Obiwan
    Obiwan

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    #181866

    Robert Butler
    Participant

     “In a one-sided or one-tailed test of a mean or of a difference between two means, the investigator pays attention only to deviations from the null hypothesis in one direction, ignoring deviations in the other direction.  This happens primarily in two situations.
    1. The investigator knows enough about the circumstances of the test to be certain that if the mean is not equal to some target/designated value then the mean is greater than the target/designated value or if the difference between the mean and the target is not 0 then the difference is greater than 0.
    2. An investigator is comparing a new treatment A with a standard treatment S with mean Xbar0 that has proved satisfactory. The investigator chooses XbarA less than equal to Xbar0 as the null hypothesis and XbarA greater than Xbar0 as the alternative hypothesis since the new treatment is of no interest unless it is superior to the standard treatment.”
    Statistical Methods – 7th Edition Snedecor and Cochran pp.67
      In other words one tailed tests assume differences are only of interest in one direction and they assume you know which way you are looking when you compute them.
      The statement “A hypothesis test is performed with null hypothesis Ho: mu = 200 and alternative hypothesis Ha: mu < 200." Would suggest the direction of interest did matter and that the interest is only for the case where the mean of the new measurements is < 200.
     
     Without that piece of information there really isn’t much one can say about the choice of testing except to ask for a better statement of purpose.

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    #181867

    Robert Butler
    Participant

    Correction – drop the last sentence – I wrote the post in a word document I was using for another purpose and copy pasted a run-on thought for something else.

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    #181869

    Obiwan
    Participant

    Robert
    I can agree with this, I have just always found it to be more correct when you add the “greater than” (in this case) to the null.  Your statement of “In other words one tailed tests assume differences are only of interest in one direction and they assume you know which way you are looking when you compute them.”…contains a lot of assumptions.  If you put the greater than statement into the null, as I suggest…no assumptions have to be made.
    In reality, I keep my hypothesis tests simple…I apply a two tailed test in most every case.  I know this effectively changes my alpha if I am only worried about one tail, but it also keeps it much simpler!
    Obiwan

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    #181871

    Robert Butler
    Participant

      In the context of the discussion concerning the choice of a one or a two sided test one isn’t making any assumptions at all – what one is (or should be) doing is demonstrating that he/she understands the problem and the focus of interest of the individuals doing the work.  As for simplicity – a default choice of a two sided test is many things but it is not simple.  If the situation calls for a one sided test and you insist on a two sided test then you are insisting on needless experimental effort- in other words you are deliberately wasting time and money.  I realize there are situations where the cost in terms of time and money per experiment/sample are so low that this isn’t much of an issue but I’ve never had the luxury of working in environments of this type.  

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    #181874

    Mikel
    Member

    You may well reject the null incorrectly too.

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    #181876

    Mikel
    Member

    Wow – you guys are scaring me. Ever read a stats book?

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    #181878

    Ken Feldman
    Participant

    Books aside. For an example, I created a random data set of mean equal to 300 and a s.d. of 1. I tested the null equal to 200 and the alternate as less than 200. The p value came back 1.0 indicating that I can’t reject the null. But it is obvious that the null is not true. How do we deal with that? Are we saying that the researcher should have known the mean of the data was 300 and tested for greater than 200 rather than stupidly going with the Ha as less than? But Belts sometimes do stupid things and draw bad conclusions. If I tested for the Ha as not equal to 200 then I would have rejected the null and then been able to see from the descriptive statistics which direction was it “not equal to”. A bit more poke yoke is what I believe Obi is trying to say. Now isn’t this more fun than talking about the 1.5 shift and training in Mumbai?

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    #181882

    Craig
    Participant

    Scary indeed. Looking up t-critical from a table is not such a major task. I blame it all on the computers that spit out p-values. We have just become plain lazy and don’t care about t-tables any more!
    Imagine if the question was about F-critical? numerator degrees of freedom, denominator degrees of freedom….the debate would last forever :-)
     

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    #181896

    Ken Feldman
    Participant

    Ahhh, the good ole days. My favorite was the phrase, “zone of rejection”. Sounded like me and the sorority girls.

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    #181903

    Obiwan
    Participant

    Darth
    Much obliged…this is exactly what I am trying to say.  My only contention…and I understand what everyone has said about my option of doing only 2 sided tests (yes, they sometimes are too rigid when you are only concerned about 1 side).
    But, my original contention is simply that, in this situation, you should put the greater than symbol in the null hypothesis.  This then covers any situation.  In the situation that Darth proposed, what would you tell an Executive (who, obviously would have no where near this level of understanding!)…that “no Sir, I realize that 300 is much larger than 200, but according to my statistical tests, they are equal!”  To me, that is a one of the reasons we often lose people.  If I said that “yes Sir, my test shows that 300 is greater than or equal to 200″…I have a much better opportunity for his understanding.
    Obiwan

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    #181963

    Robert Butler
    Participant

      I have to admit I’m surprised at the content of several of the posts to this thread and I’m also very concerned.  Consequently I spent the better part of last evening trying to compose a response that I hope will be informative, critical, and non-incendiary.
     
      Rather than putting books aside let’s do a recap.
     
     
     The original poster said:
     
    “A hypothesis test is performed with null hypothesis Ho: mu = 200 and alternative hypothesis Ha: mu < 200. Alpha should be 0.01.
    I am trying to find out the right t-value. Because I have 12 values, the degrees of freedom should be 11. Now my question: It is a one tail test so that I need to look for Alpha (0,01; 11 degrees of freedom) = 2.718? That is what I think.
    However the solution should be another value: 3.106.”
     
    I, and the book I cited, said:
     
    “In a one-sided or one-tailed test of a mean or of a difference between two means, the investigator pays attention only to deviations from the null hypothesis in one direction, ignoring deviations in the other direction.  This happens primarily in two situations…..”
     
    …and the key issue in the two situations was the fact that direction of inequality mattered – in other words a difference in one direction was of interest and there was no interest in a difference in the opposite direction.
     
     
    If you are going to successfully attack a problem you need to know its definition and you need to know what does and does not matter.  In every statistics text and in every situation where the need for a two sided test is indicated the words describing the problem are of the form
     
      “is the mean of the sample equal to the target/mean of the second sample/our historic mean/etc.?”
     
    Or
     
      “is the mean of the sample different from the target/mean of the second sample/our historic mean/etc.?”
     
      Please note that no direction is specified. 
     
      On the other hand, when direction does matter the words describing the problem are of the form
     
      “is the mean of the sample less than the target/mean of the second sample/our historic mean/etc.”
     
    Or
     
      “is the mean of the sample greater than target/mean of the second sample/our historic mean/etc.”
     
       If symbols such as > and < are used in their stead then these too are the hallmarks of direction preference.
     
    Part of your job as an analyst/BB/statistician/problem solver is to ask questions that will clarify and properly define the problem.  I always ask about the issue of direction and if, in fact, direction does matter and there is no interest in the alternative then I use a one tailed test since that is the correct way to approach the problem.
     
    To that end, if I had a situation where less was better and more was of no interest and we put together a test where the focus was on whether or not the sample mean for the new process was < 200 then, if after running the tests, the sample mean for the new process turned out to be 300 there would be nothing to test. 
     
    There would be an expression of shock and amazement on the part of my investigators for the simple reason that we would never have run the tests if we had expected to see something that was obviously worse and there would be a huge post mortem with respect to checking all aspects of the data collection process and prior work that lead us to the belief that we could reasonably expect a result <200  but no one I’ve ever worked with would ever ask me to test to see if 300 was significantly less than 200 – note I didn’t say “different” the issue, as framed in the investigation, and as understood by everyone (including that brain dead exec from the earlier post), was not about “different” it was about “less”.
     
      If you as a BB/analyst/statistician/man-woman on the spot ignore or refuse to determine the intent of an investigation and choose instead to make one-size-fits-all statements such as “(yes, they sometimes are too rigid when you are only concerned about 1 side).”  Then what you are doing is refusing to do the job you were paid to do and you are attempting to use language to turn your refusal to work/think into a virtue.
     Rather than generate poorly framed theoretical situations let’s look at a real one.
    Three months ago one of my investigators had the chance to get in house funding for some exploratory research.  The monetary grants for these efforts are controlled right down to the last penny.  I had worked with her on several projects that were peripheral to the work she was doing and in each of those cases the dictates of the problem required sample size estimates based on the assumption of a two sided test.  I met with one of her co-workers to discuss the writing of my part of the proposal and I asked about testing and was assured we were again going to use two sided.  About a month later she called a meeting about the proposal- the issue was very simple – her plan was $1424.37 over the maximum allowed.  Her choices – lose the amount or lose the chance to do the work. 
    All of us sat down and began talking cost reduction.  In the course of the discussion she said “I really think this new procedure will be much more effective than the current one.”  I interrupted and asked if this meant the focus of the new work was only on improvement.  Her response was in the affirmative – there was no interest in a lookalike or in something that was worse.  My response was – “Well, I know that the cost per sample for prep and analysis is $1505.00 and I know that for a one sided test the sample size is going to go down by at least one – looks to me like we have your cost savings.”  I was asked to re-run the calculations and it turned out that instead of 26 samples for a two sided test we only needed 24 for a one sided test.  The rewrite was done – the proposal was recently accepted and work will begin shortly. 
    In this case and in others I’ve worked on the virtue of  “covering any situation” when in fact any situation was not what was needed would very likely have condemned her proposal to the scrap pile or perhaps needlessly weakened it because we tossed out some other aspect of the proposal in order to meet the monetary target.
     At the end of all of this the person for whom I have the most sympathy is the original poster.  The exam question he/she provided does ask a question framed such that direction is a factor.  Consequently the t statistics should be, as he/she noted, 2.718.  That the “correct” answer is given as 3.106 highlights the fact that the individual writing the question doesn’t really know how to write a test question about one/two tailed tests. This puts the original poster in a nightmare situation– he/she could easily fail the exam not because he/she didn’t understand the material but because the person who wrote the exam doesn’t know what they are doing.

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    #181986

    Ken Feldman
    Participant

    Robert, as usual, a well thought out but wordy response. Despite the many views expressed in the thread, I believe we have all come to the conclusion that doing a one sided test is good, more powerful and the researcher needs to understand whether that is important and what direction is of interest. Obi’s approach was a workaround for idiots who probably shouldn’t be doing the analysis anyway. Thanks for putting the effort into your post.

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