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Suppose it is required to test the fitting of binomial distribution to the following data ( at level $\alpha =0.05$) :

enter image description here

the parameter p is estimated as p = 0.494 , and the expected frequencies (given in red ) are also calculated .

let the claculated value of chisquare statistic (using the formula $\sum\frac{(O_i - E_i)^2}{E_i}$ ) be $\chi_{cal}^2 $ (say)

my question is : what type of hypotheses does "goodness of fit" signifies , i.e is it single tailed or two tailed ?

in other words ( or expressing my question more clearly by showing the way i attempt it)

$ H_0 : $ $ P = 0.494 $ or fit of the model is good

$ H_1 : $ $ P \ne 0.494$ or fit of the model is not good

then the test criteria should be to accept $ H_0 $ iff

$\chi_{0.025}^2 \le \chi_{cal}^2 \le \chi_{0.975}^2 $

but thats not the actual story , every source that i reffered , says that test criteria should be to accept $ H_0 $ iff

$ \chi_{cal}^2 \le \chi_{0.05}^2 $ , why is this happening , if its a two tailed hypotheses then both ends should be reffered ? please correct me where i am wrong

ANUJ NAIN
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  • I explain this in great detail (but nontechnically) at http://stats.stackexchange.com/questions/31/what-is-the-meaning-of-p-values-and-t-values-in-statistical-tests/130772?s=2|0.0000#130772. You might also be interested in the thread at http://stats.stackexchange.com/questions/4360. – whuber Apr 07 '17 at 18:07
  • Although you are literally looking at only one tail of the Chi Square distribution, you are testing a two-tailed hypothesis in that you would reject the null if your observed frequencies were too high or too low. – David Lane Apr 07 '17 at 18:08
  • @David Although that's correct, I suspect your characterization might confuse the issue more than clarify it. The reason is that the "tail" of any hypothesis refers to the distribution of the *statistic,* whereas in this case the observed frequencies are not the statistic being tested. – whuber Apr 07 '17 at 18:50
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    @ whuber. That could be although the test statistic is derived from differences between expected and observed frequencies regardless of direction. We would be better off with the phrase "one-directional" rather than "one tailed" . A lot of students are confused by an F test where one tail is used but it is not a one-directional hypothesis. – David Lane Apr 07 '17 at 21:07
  • @whuber I apologize to say but the link you provided me is BIBLE to read ,please can you be specific for this question and then explain me here , – ANUJ NAIN Apr 08 '17 at 05:23

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