What causes the null hypothesis to be rejected?

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    This is a roll up your sleeves, sharpen your pencil, adjust your pocket protector kind of a question…Instead of putting out a disclaimer, I’ll get on with the question:
    In the t-test, for significance of a relationship between a independent variable x and dependent variable y, you must reject the null hypothesis by showing that B1 (the slope of the linear equation) does not equal zero.  To do this t, defined as t=b1/Sb1, where b1 is the slope and Sb1 is the estimated standard deviation of b1, must be greater than talpha/2, where t alpha/2 is the t distribution with n-2 degrees of freedom (we assume alpha to be .01), or t must be less than -(alpha/2).
    talpha/2 is usually a fairly small number (when a reasonable data set-n- is used and degrees of freedom equals n-2).  This obviously means in order for us to reject the null hypothesis, t must be a strong positive number(which it will be as long as the linear equation has a positive slope) or negative number.
    My question, besides needing b1 to be large positive or negative, what else would cause us to reject the null hypothesis?  That is, what makes this slope large enough or small enough to reject the null hypothesis?
    b1 = ( (Exiyi) – (ExiEyi)/n )/( (Exi^2)-(Exi)^2 )/n )
    Exiyi – summation of all xs*all ys
    ExiEyi – summation of all xs * summation of all ys
    Exi^2 – summation of all xs squared
    (Exi)^2 – summation of all xs, then square that number (much bigger usually then Exi^2
    n= number of data points
    Clear as mud?  Thanks in advance.



    I am uncertain of the exact nature of your question.  You appear to be discussing the one sample t test for the significance of the estimated slope in a simple linear regression analysis.  The usual null hypothesis is that that the true underlying slope value is 0 (no relationship exists between Y and X).  Under a mild assumption of normality (the t test works well for data that deviate considerably from the normal) and the null hypothesis, the ratio of the slope estimate b1 to its associated standar error S(b1) is known to follow student’s t distribution with n-2 degrees of freedom.  For a given alpha, say 0.05, one looks up the (1-alpha/2) percentile of the student t distribution — the alpha/2 percentile is just the negative of the upper tail percentile.  If the computed t ratio is below the lower percentile (t _n-2, alpha/2)or above the upper percentile (t_n-2, 1-alpha/2), then we have evidence that the null hypothesis is incorrect. If the null is false, then the distance between the estimated slope b1 and the null value of 0 is too great (in units of standard error) . If the null hypothesis is false then the ratio of b1 to S(b1) follows the noncentral t distribution and is the basis for the power calculations that can be done for the test.  I do not know if this was the point of your question or not, but I thought I would attempt an answer.


    Andrew M. Brody

    Boy, you’ve got me by the throat.  I’m too far away from this.  Does E=MC2 ring a bell?
    Hoping you have a sense of humor,
    Andy Brody



    Keeping off the maths I think the basic answers to what will guarantee a low p-value are:
    1. The difference in you X settings – If you have a process that is really sensitive to temperature but you only test at say 50 degrees and 60 degrees then the maths cannot see a difference. If you test at 50 degrees and 150 degrees this may create a significant difference.
    2. The amount of noise around the model – If when you look at a scatter diagram all the points are on the line or close to the line, the hypothesis test will spot smaller changes.
    Before you attempt to construct a model you should consider very carefully am I testing over the correct range (10 degrees or 100 degrees) and is the process sufficiently in control for the maths to see signals above the noise.
    Hope this helps


    Mike Nellis

    When these basic rules below are comprimised..the t-test will be affected:
    1.  both samples should be from independant populations described by a normal distribution…if not, normal try transformations
    2.  the standard deviations or variances of each sample are equal…use normality tests
    3.  and as always, sample size & measurement system error can also effect your t-test
    p-values could further support rejecting the null hypothesis, Ho
    hope this helps,


    Sam M

    Here is a good one for rejecting the null
    When p is lo, Ho must go.  where p = probability and Ho is the null hypothesis.


    Mark Volz

    Another approach is to do a correlation analysis between your Xs and Ys. The null hypothesis is that there is no linear relationship. You would need the analysis to produce a low p-value, below your predetermined threshhold to reject the null hypothesis and conclude there is a linear relationship. You can use Minitab or any other tool that can calculate correlation. Good Luck!



    HiAlso, look at following parametersp value (u can go they told me upto 0.1 for regression analysis, but not higher than 0.05 for hypothesis testing)r square value – preferably more than 0.70 (when you have some measurement and noise issue)look at scatter diagram conduct lack of fit test (MINITAB will do it for u in a spiff)having done that you can safely assume that there is a relation or no relation b/n Y and X…hope this was helpful



    Thanks all for your help!
    I guess what I was looking for isn’t really attainable –  a simple way to say ‘if the third x value in a data set of n points is 3.247 units away from the linear regression line, then we must fail to reject the null hypothesis’.  This type of ‘easy’ way of looking at the relationship between x and y just doesn’t exist, and that must be the reason why all of the statisical equations were developed in the first place. 
    After all of the posts and thinking about the relationship for a while, the basic t-test equation is the easiest way to explain it: if tt_alpha/2, then we reject the null hypothesis in a t-test for significance. 
    If anyone else is reading this post and it isn’t making sense, I would just suggest sticking to a high level view by plotting your data and looking at the graph.  Insert very large/small values that causes failure to reject he null hypothesis, and then insert data that allows rejection of the null hypothesis – the difference is obvious.
    Thanks again to everyone for their input!

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